A Robust Neighborhood Truncation Approach to Estimation of Integrated
Quarticity

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Abstract:

We provide a first in-depth look at robust estimation of
integrated quarticity (IQ) based on high frequency data. IQ is the
key ingredient enabling inference about volatility and the presence
of jumps in financial time series and is thus of considerable
interest in applications. We document the significant empirical
challenges for IQ estimation posed by commonly encountered data
imperfections and set forth three complementary approaches for
improving IQ based inference. First, we show that many common
deviations from the jump diffusive null can be dealt with by a
novel filtering scheme that generalizes truncation of individual
returns to truncation of arbitrary functionals on return blocks.
Second, we propose a new family of efficient robust neighborhood
truncation (RNT) estimators for integrated power variation based on
order statistics of a set of unbiased local power variation
estimators on a block of returns. Third, we find that ratio-based
inference, originally proposed in this context by Barndorff-Nielsen
and Shephard (2002), has desirable robustness properties in the
face of regularly occurring data imperfections and thus is well
suited for empirical applications. We confirm that the proposed
filtering scheme and the RNT estimators perform well in our
extensive simulation designs and in an application to the
individual Dow Jones 30 stocks.

1 Introduction

Important progress in measuring and forecasting return
volatility has been obtained through techniques exploiting the
information in intraday price movements. The use of high-frequency
data is, however, not without its problems. The main complication
is the pronounced inhomogeneity of the intraday return series as
diurnal patterns interspersed with news events and market
microstructure frictions complicate direct modeling of the high
frequency dynamics and introduce a variety of idiosyncratic
features that are largely irrelevant for inference about
inter-daily volatility. The realized volatility (RV) approach
"solves" this problem by aggregating the intraday return
observations to a daily frequency in a manner that retains the
majority of the inherent volatility information while mitigating
the impact of noise and diurnal patterns. The RV approach has been
widely adopted ever since its formal introduction as a
nonparametric estimator of the return variation in Andersen and
Bollerslev (1998).1 In parallel, a large body of
theoretical work on model-free estimation and inference for
components of the realized return variation process has arisen.
Initial econometric issues are addressed in Andersen, Bollerslev,
Diebold and Labys (2001, 2003) and Barndorff-Nielsen and Shephard
(henceforth BNS) (2002).2

Conceptually, realized volatility differs from the standard
notion of volatility by focusing on ex-post measurement of the
realization of the (stochastic) return variation rather than the
(ex-ante) return variance. Once attention shifts to the actual
volatility realizations, new questions arise. For example, how do
we assess the accuracy of our (daily) ex-post measures of the
integrated return variation and how do we identify the impact of
jump components. Such features are critical for a variety of issues
in real-time financial management, including volatility
forecasting, analysis of the dynamic properties of jumps and news
events, derivatives pricing, estimation of return correlations,
determination of return-volatility asymmetries (the leverage
effect), and developing insights into the interplay between return
volatility and the macroeconomic environment.

The key ingredient for inference regarding the return variation
and the presence of jumps is the so-called integrated quarticity
(IQ). To illustrate the importance of accurate IQ measures we
review a few results from the RV literature. We denote the
continuously evolving log-price for a financial asset by
. Under general conditions, the log-price
constitutes a semi-martingale with respect to an underlying
filtered probability space. The associated ex-post realized
quadratic variation, , for over
may be decomposed into an integrated
(diffusive) volatility, , and a residual (jump)
component, ,

(1)

where and denote the
instantaneous drift and diffusion coefficients, while and are adapted Wiener and finite
activity jump processes, respectively.

For a given trading day,
, we consider the ideal scenario
in which we observe equally-spaced (log)
returns,
,
. In this case, the
realized volatility (RV) is a consistent nonparametric estimator of
QV, as the number of intraday observations diverges,
(in-fill
asymptotics),

where
,
which, as observed by BNS (2002), can be consistently estimated
from the high-frequency data themselves via the Realized Quarticity
(RQ) statistic:

Clearly, accurate inference about the integrated variance hinges
on reliable estimates for IQ. Unfortunately, IQ estimation is
challenging. It involves estimating fourth order return moments
from noisy intraday return series impacted by the confounding
effects of market microstructure frictions, diurnal patterns,
outliers, and other data irregularities. For example, it is well
known that the RQ estimator is highly imprecise and non-robust to
such features, even if jumps are absent. Moreover, when discrete
price changes do occur, RV is no longer consistent for IV, and the
RQ statistic diverges:
as
. Given the
compelling evidence for jumps, this is critical in practice. In
response, various jump-robust IQ estimators have been developed,
but they are subject to potentially serious finite sample biases.
At present, there simply is no systematic evidence regarding the
performance of alternative jump-robust procedures for empirically
realistic scenarios.

Recognizing these issues, a variety of ad hoc IQ estimation
procedures have been implemented in the empirical literature.
Before the jump-robust theory was developed, the RQ statistic was
used, but only with relatively coarse sampling. For example, BNS
(2004a) exploit 10-minute foreign exchange returns, while Bandi and
Russell (2008) recommend computing RQ from 15- or 20-minute
returns, as sparse sampling mitigates the impact of outliers and
microstructure noise. Later, BNS (2004b) and Huang and Tauchen
(2005) rely on 5-minute returns for constructing jump-robust
estimators of IQ.3 Finally, due to the distortions
arising from market microstructure effects, Jiang and Oomen (2008)
opt for simply squaring their jump-robust IV estimator to obtain an
IQ estimator, thus settling for a substantial Jensen inequality
bias, but aiming to reduce estimation uncertainty.

To illustrate the practical importance of jump robustness,
consider drawing inference about the IV of IBM stock returns across
three days in February 2008 using the non-jump robust RQ/RV
measures versus a pair of jump robust measures, as shown on Figure 1.4

We plot prices (blue line),
the IV point estimate (red line), the inter-quartile range (blue
box) as well as two standard deviation IV confidence bands (black
whiskers) for IBM for three trading days in February
2008.

The jump on 2/26/2008 is readily identified visually and easily
detected using a jump robust test statistic. In fact, the robust
MedRV estimates and associated standard error bands, based on
MedRQ, suggest a relative stable volatility process across the
three trading days. In contrast, the regular RV estimate for IV is
greatly inflated on 2/26/2008, and the confidence band is huge,
reflecting a diverging RQ statistic. Hence, the reliance on non
jump-robust statistics has two consequences. First, when jumps are
present the IV estimate is upward biased because the jump component
in QV is attributed to IV.5 Second, the associated confidence band
is grossly overstated, indicating very poor estimation precision
whereas, in fact, the robust estimate appears quite reliable.
Hence, non-robust inference may produce excessively erratic IV
estimates and convey a sense of exaggerated imprecision associated
with these techniques. While the misleading inference afforded by
the regular RV and RQ estimators is apparent in Figure 1, at least when
contrasted with the robust inference and a depiction of the price
path, it can be less obvious in cases with higher volatility levels
and relatively smaller jumps. As such, it is important to develop
feasible robust and efficient procedures for estimating IQ and
conducting inference for IV.

One main contribution of this paper is to provide a first
in-depth exploration of the virtues and drawbacks of alternative
jump-robust estimation procedures for IQ, including their
robustness to a variety of realistic features of the return
generating process. A point of emphasis is the use of wide
pre-averaging windows for controlling the impact of microstructure
noise on the inference. This enhances robustness and simplifies the
distribution theory as the impact of noise is annihilated
asymptotically. A second contribution is the development of a new
class of robust neighborhood truncation (RNT) estimators
that generalize existing nearest neighbor and Quantile RV
estimators. They involve the application of a second layer of order
statistics to suitably chosen return functionals, thus robustifying
the inference for IQ with only a minor loss of efficiency. We find
such RNT estimators to perform admirably, especially when used in
combination with the ratio statistic, , which
is known to provide improved finite sample inference for IV.
Moreover, these principles apply generally and can be used to
enhance the robustness of inference from alternative classes of
estimators. A third novelty is the use of an outlier filtering
procedure that operates directly on an estimation functional of
interest rather than on individual returns. This functional
filtering principle adapts the filter to the specific
assumptions underlying a given estimator. Hence, it controls the
impact, and potential distortion, of abnormal outliers within the
exact metric in which they contribute to the ultimate estimator. In
applications to individual equity return data we find this filter
indispensable for rendering entire classes of promising candidate
IQ estimators viable. The unifying theme behind our new estimators
and universal filtering procedure is to operate directly on the
functional space of local power variation estimates rather than the
individual returns. Nonetheless, the latter, and common, approach
may be obtained as a special case of our procedure.

The remainder of the paper is structured as follows. Section 2
reviews the modern approach to robust estimation of integrated
power variation. Section 3 develops our robust neighborhood
truncation estimators. In Section 4, we discuss additional
procedures applied to obtain robustness against jumps and noise.
Section 5 illustrates the importance of common data features for IQ
inference through an extensive simulation study. Finally, Section 6
provides evidence using high-frequency returns on the Dow Jones 30
stocks, while Section 7 concludes. All proofs are relegated to the
Appendix.

2 Overview of Jump-Robust Power
Variation Estimation

This section summarizes the modern approach to power variation
estimation. We outline the theoretical setting and review some
existing estimators which are later used in our simulation study
and empirical investigation. In the process, we discuss practical
trade-offs that must be confronted in estimating objects involving
high powers of volatility.

2.1 The Theoretical Setting

We focus on a single asset traded continuously in a frictionless
market over the period , referred to as a
trading day. If it is a limited-liability asset with an expected
positive payoff at some future date, the price will remain strictly
positive. No-arbitrage conditions then ensure that the log-price
process constitutes a semimartingale with respect to the underlying
filtered probability space, see, e.g., Back (1991) and Andersen,
Bollerslev and Diebold (2010). Hence, for most of our analysis we
invoke the following conditions.

Assumption 1 The continuously compounded return
process, is governed by a jump-diffusive
semimartingale,

(2)

where is a locally bounded and predictable
process, is an adapted cadlag process bounded
away from zero, and is a finite activity jump
process.

where is locally bounded and predictable,
are cadlag, the
Brownian motions are uncorrelated, and
is a finite activity jump
process.

If the Brownian component in Assumption 1 is non-zero, the
return innovation is an order of magnitude larger than the expected
return over short time intervals, implying that the drift term
typically does not affect the asymptotic distribution of power
variation estimators based on high-frequency data. Hence, we ignore
the drift term in this section.6

Another key implication of Assumptions 1 and 1A is that we may
derive the asymptotic properties of many relevant estimators
assuming that the intraday returns are locally Gaussian. To
operationalize this approach, the trading period is broken into
smaller blocks. For each block, we treat
volatility as constant, even if the actual return variation evolves
stochastically and the price path contains finite activity jumps.
If equally-spaced continuously compounded
returns are available, and each block contains returns, we assume, without loss of generality, that
. Notice that each block covers
of the trading
period and each return reflects the price evolution over an
interval length of
.

The above insight simplifies matters greatly, as nonparametric
jump-robust estimators now are easy to devise. One simply selects a
suitable unbiased estimator for the (power of) volatility within
each block under the null hypothesis of i.i.d. Gaussian returns,
and then cumulate the estimates across blocks to obtain the overall
power variation. The distribution theory is developed using
standard in-fill asymptotics, letting grow
indefinitely, while requiring
. In most cases,
is fixed and diverges
proportionally with .

2.2 Estimating Power Variation under the
Diffusive Null Hypothesis

We first consider the case where there is no jump component.
Given the assumptions invoked above, we focus on the null
hypothesis that the returns within a small block are i.i.d.
Gaussian. A generic estimator of the
order return variation, for
an even positive integer, is now obtained
as the average of local estimates of
based on a functional
operating on blocks of
adjacent returns. For each integer
, we have a return
block,
. Under the null, these returns are i.i.d.
. We
let
denote the
functional exploited by a given estimator to obtain an unbiased
estimate of
for the 'th block. If Assumption 1 holds, the power variation
estimator is consistent. Heuristically, the law of large numbers
implies, as
,

A corresponding central limit theory may typically be devised if we
invoke Assumption 1A.

The simplest estimator within this framework is the realized
Power Variation (PV) measure. It does not exploit multi-return
blocks, so . It takes the form,

where
for

The normalization constant is given by
, for any .

For , this produces the regular RV, or
PV(2), estimator with ,
while yields the RQ, or PV(4), estimator from Section 1 with normalizing constant
.

A couple of comments are warranted. First, the setting ignores
data errors and market microstructure frictions. Higher order
return moments are particularly sensitive to faulty price
observations or inappropriate assumptions regarding the evolution
of the high frequency returns. We discuss these issues in the
context of the simulation and empirical sections below. Second, in
contrast to the realized power variation estimator, the functional
will in the following be designed to be
jump-robust, i.e., provide valid asymptotic inference for the power
variation, even in the presence of finite activity jumps. However,
jumps often have a severe adverse effect on the finite sample
properties of the estimators, especially for . Many of the practical complications below arise
from this feature.

2.3 Jump-Robust Power Variation
Estimation

We now outline the basic principles behind the construction of
power variation estimators that are robust to the presence of
finite activity jump processes. Asymptotically, as the block sizes
shrink towards zero and the number of blocks grows indefinitely,
there will be a finite number of blocks containing one single jump
each. Hence, in the limit, the power variation associated with the
blocks containing jumps is negligible. It follows that the power
variation can be estimated consistently as long as the contribution
from the "jump blocks" is an order of magnitude less than the
overall power variation measure which, of course, is . However, the jumps are also of order , so the functional must ensure that
the jumps are dampened sufficiently to eliminate their impact
asymptotically.

Formally, for any given sampling frequency, we denote the set of
indices corresponding to returns for which the associated block
contains a jump by . Thus, for , there is a jump in the return block
. We then write the
generic power variation estimator as,

The first term estimates the integrated power variation
consistently, i.e.,

as

The contribution from the blocks containing jumps is negligible,
in the limit, only if each such block is of order less than
. Thus, the associated power variation
estimator is consistent as long as
for
.7. This is accomplished
in different ways by alternative jump-robust estimators. Moreover,
their practical effectiveness is largely determined by the degree
to which they accomplish sufficient dampening of the jump
contributions in finite samples.

2.4 Alternative Jump-Robust Power
Variation Estimators

2.4.1 Multi-Power Variation
Estimators

The first (finite activity) jump-robust power variation
estimators were the Realized Multi-Power Variation (MPV)
statistics, inspired by BNS (2002). Expressed in terms of the
functional applied to successive return blocks, the estimator takes
the form,

For and or , the estimator reduces to the (non jump-robust) RV or RQ
estimator, respectively. Prominent (jump robust) special cases
include
, which defines the bipower
variation statistic, and various IQ estimators, such as tripower
, quadpower
, and quintpower
.

As described earlier, the actual estimator is now obtained by
averaging the value of the functional across the available
blocks,

The MPV estimator is consistent and affords an associated CLT,
as long as is chosen sufficiently large relative
to . This produces an inevitable bias-variance
trade-off. A larger implies more dampening of
the jump term, so the finite sample bias induced by the jump is
alleviated. On the other hand, for a given sampling frequency, a
larger block size, , implies that the
functional is less localized, so the constant volatility assumption
provides a poorer approximation, and the estimator becomes less
efficient.

2.4.2 Truncated Power Variation
Estimators

An estimator closely related to the PV and MPV statistics is the
Realized Truncated Power Variation (TPV) measure. Mancini (2009)
introduces the threshold realized volatility and quarticity
estimators, while Corsi, Pirino and Reno (2010), henceforth CPR,
consider a bipower variant of these statistics. These estimators
achieve jump robustness by truncating observations exceeding a
pre-specified threshold. Under in-fill asymptotics, we may
stipulate that the threshold converges toward zero slowly enough
(slower than
) that the limiting
distribution of the resulting estimators is identical to their
non-truncated counterparts. In particular, Truncated RV, or TRV, is
asymptotically most efficient among all jump robust IV estimators,
and similarly the Truncated RQ, or TRQ, is the most efficient
jump-robust estimator for IQ. Moreover, it is evident that the
(finite sample) jump distortion is determined by the size of the
truncation threshold and thus is under direct control in designing
the estimator. The block-functional defining the truncation
multi-power variation estimator of order with
truncation threshold,
, takes the
form,

where
is an indicator function,
taking the value of one if the statement A is true, and zero
otherwise. As before, the actual TPV(m,p)
estimator is obtained by averaging the functional values across the
available return blocks for the trading period.

The choice of threshold can be delicate. It is beneficial to
truncate aggressively to reduce the jump distortion by choosing a
low threshold, but the non-jump returns are then also truncated
with non-trivial probability. CPR suggest a finite-sample scaling
to correct for this bias. They develop an iterative scheme aiming
to obtain a fixed point at which the expectation of the truncated
estimator equals the true (estimated) volatility under the null
hypothesis. The approach is conceptually appealing, but has
drawbacks. First, the modified estimator is no longer linear in the
unobserved
and thus suffers from a downwards
bias, due to Jensen's inequality, even in the ideal Brownian case.
Second, they use a sizeable two-sided window (e.g., 50
observations) to obtain a local volatility estimate, thereby
rendering it susceptible to an additional bias due to time
variation in volatility across the block.

2.4.3 Neighborhood Truncation Estimators

Andersen, Dobrev and Schaumburg, henceforth ADS, (2012)
introduce a couple of IV estimators, MinRV and MedRV, designed to
improve on the trade-off between jump robustness and efficiency
confronting the MPV estimators. MinRV and MedRV are based on an
endogenous "nearest neighbor" truncation which is particularly
helpful in alleviating the finite-sample impact of isolated large
jumps. We now extend this theory to cover general power variation
estimation. We start out by introducing notation that allows us to
identify various order statistics associated with a given return
block.

First, we denote the
th block, consisting of absolute returns
raised to the
power, by
Next,
indicates the
order statistic of the block
, so
. As the returns are assumed
we may
also write
, highlighting the fact that all estimators for the block
ultimately are functionals operating on the realization of an
-dimensional standard normal random
vector.

We now readily obtain separate unbiased
estimators for
, namely one for each order
statistic. We denote these "Neighborhood Truncation" estimators,
or NT(j,m,p). As before, we construct them by
averaging the appropriate block functional across the trading
period. The functional takes the form,

where
This normalization ensures that the functional provides an
unbiased estimator for
. Since the scaling factors
are inversely related to the expected value of the order
statistics, we have the ranking,
. For
high values of and , these
factors become quite large for the lower order statistics, while
they are very small for the higher order statistics.8

The class of neighborhood truncation estimators generalize the
MinRV and MedRV estimators, as we have,
and
.9 Under appropriate
conditions the NT estimators are consistent and afford a CLT. We
confirm this result when we introduce an even wider class of robust
estimators in Section 3.

2.4.4 Combining Power Variation
Estimators

It seems natural to combine some of the estimators introduced
above to obtain superior asymptotic properties and, possibly,
improved finite-sample performance. It is, however, outside the
scope of the current paper to pursue this topic in depth.
Nonetheless, we do develop the framework and notation to
accommodate such combination estimators, as it is useful for our
introduction of a new class of robust estimators in the next
section.

Assume we have a candidate set of separate
jump-robust power variation estimators which all are unbiased,
consistent and afford a CLT under the local Gaussian null
hypothesis. We denote this set of estimators,
Almost trivially, it is then, in theory, feasible to improve the
performance of any single estimator by combining it with
others.10

We formalize the selection of a subset of the estimators by
introducing a "selection" vector, identifying the elements in
used to construct a given
(combination) estimator. Hence, we let the x vector
consisting of an ordered subset of integers from
indicate
that the combination estimator is based on the set
. Denoting the set of all possible selector vectors
, any subset is now uniquely
identified by
, where
ranges from a scalar
(using a single estimator), at the one extreme, to the full vector
(using them
all), at the other extreme.

A natural way to preserve the desirable properties of the
individual estimators is to exploit linear combinations with
non-negative weights that sum to unity. For example, focusing
exclusively on a set of NT estimators constructed using a return
block of size , we have
, where each element represents an unbiased estimator based on the
corresponding (absolute return) order statistic. Picking a specific
NT estimator amounts to, a priori, selecting a given integer
There
are a total of distinct non-empty subsets
of
from which to construct a
combination estimator. It is a routine exercise to extend our
asymptotic results to cover the case of any such linear combination
of NT estimators.11 Conceptually, it is likewise
straightforward to derive corresponding results for linear
combinations involving alternative types of jump-robust
estimators.

In summary, within the ideal setting of Assumptions 1 and 1A,
superior asymptotic performance can be obtained by combining the
information associated with all available estimators. However, this
must be weighed against the robustness objective of ensuring
reliable finite sample inference in the presence of jumps as well
as other potential sources of noise. Such robustness concerns
motivate the introduction of an even broader class of combination
estimators in the following section, obtained by nonlinearly
combining suitably chosen unbiased estimators via a second layer of
order statistics.

3 Robust Neighborhood Truncation Estimation

Our major objective is to develop a reliable jump-robust
procedure for estimating measures associated with the integrated
quarticity. Most of the estimators reviewed in the previous section
were developed for IV, even if they can be adapted for higher order
power variation measures. It is worth recognizing that the relative
importance of factors impacting the trade-off between statistical
efficiency and robustness changes substantially as we estimate
higher order power variation measures. In fact, our simulation
evidence demonstrates, quite strikingly, that the most suitable
approach for IV estimation is unlikely to be preferable when
estimating IQ. Consequently, we now introduce a novel inference
procedure which enhances the robustness to common sources of
finite-sample distortions and allows for a great deal of
flexibility in implementation so that the estimator can be tailored
to the specific features of a given return series and market
environment.

3.1 Theory

This section proposes estimating integrated power variation via
a nonlinear combination of existing unbiased estimators,
obtained by invoking an additional layer of order statistics. We
develop the theory for neighborhood truncation estimators, but the
principles apply for any set of unbiased estimators. The emphasis
is on finite sample robustness to microstructure noise and jumps,
so we label them "Robust Neighborhood Truncation," or RNT,
estimators.

For a return block of size , there are
distinct NT estimators, namely one for
each order statistic. One may combine any subset of these to
produce an estimator that exploits more sampling information than
can be utilized by any individual one. The set of alternative
selections is the set,
, of non-empty subsets of
. A specific
choice is given by
The corresponding NT estimators are defined via the functionals
they apply to the underlying return blocks. To facilitate the
exposition, we use the following short-hand notation for these
functionals, applied to the
return block,

The rationale behind the NT estimators is to alleviate the
impact of extreme returns - large or small - which may be
incompatible with the i.i.d. Gaussian assumption. The robust
neighborhood truncation principle takes the reasoning one step
further by producing an estimator for
based on a suitable order
statistic among the subset of selected NT estimators. Formally, we
have,

where

Normalization is required, even if each NT estimator is
individually unbiased, because selection conditional on
observed realizations induces a bias. This is corrected by scaling
with (the inverse of) the expected value of the corresponding order
statistic for a standard normal x
vector. This normalization factor is not available in closed form,
but may be determined, to any degree of accuracy, by numerical
integration or simulation.

As before, the actual estimator is obtained by averaging the
estimates across all blocks in the trading periods, so we have,

Notice that the RNT procedure involves two layers of order
statistics: we first construct consistent NT estimators from the
order statistics of a block of absolute returns (which readily may
be extended to any set of consistent estimators), and then obtain
the RNT estimator from another order statistic applied to a subset
of these NT estimators. This provides a great deal of flexibility
in alleviating the impact of extreme returns. In line with the
logic behind the MinRV and MedRV measures, the RNT estimator is
consistent if we exclude the largest order statistic from
. Asymptotically, this ensures that
none of the NT estimators are generated from a (scaled) jump
return. Alternatively, this is also guaranteed if we avoid
constructing the RNT estimator from the largest realization of the
NT estimators, i.e., in the second
step.

Proposition 1Let a family of distinct NT estimators,
indexed by , be generated from absolute
return blocks of size . The largest order
statistic used for constructing any of these NT estimators is
denoted ,
. Next, consider the RNT estimator
obtained from the
order statistic applied to
this family of NT estimators, .

If (i) Assumption 1 holds; (ii)
and/or ; and (iii)
is a positive, even integer; then, as

If, in addition, Assumptions 1A applies, we obtain, for
a known
constant,

The proposition warrants a few comments. First, the distributional
convergence is stable with a mixed Gaussian limit, i.e., a normal
distribution conditional on the realization of the integrated power
variation,
,
where, importantly, the limiting normal variate is independent of
the (random) power variation process.12 Second, the
convergence result is qualitatively similar to those established
for existing power variation estimators, with the "efficiency
factor"
determining the
relative asymptotic efficiency of the estimator. Third, the main
objective is not efficiency per se, but good performance along with
(finite sample) robustness to jumps, noise, and other data
irregularities. Fourth, the results apply for the Neighborhood
Truncation and Nearest Neighbor Truncation estimators, as these
constitute special cases involving a particular choice for the
vector I. Fifth, the results are likely to extend to
the infinite activity jump case, given suitably tight constraint on
the size of the associated (jump) activity index.13

3.2 Illustration

The simulation and empirical work in the following sections
exploit fairly small blocks of in order to
retain resiliency relative to rapidly changing volatility levels
during the trading day. In addition, we find it useful to eliminate
estimators that stem from the lowest order statistics of the
absolute returns as these are relatively more affected by market
microstructure noise such as price discreteness and bid-ask bounce.
This is a particular concern, because these estimates are
bias-corrected by scaling the original small returns by a large
factor, implying that microstructure distortions may be amplified.
Likewise, we typically satisfy the formal constraint on the order
statistics by picking , so we avoid basing
the RNT estimator on the largest realization among the relevant NT
estimators.

For a -dimensional return block, the construction
of the
and
estimators
are exemplified in Figure 2. The
notation becomes quite involved so, for brevity, we refer to the
two estimators in the diagram as and , respectively.14 Both
play a significant role in our subsequent exposition. For these
estimators, the two smallest absolute returns are discarded, while
the remaining three are used to compute the corresponding NT
estimators. Among those, we pick the lowest, respectively median,
realization and scale it to construct the associated RNT
estimator.

Figure 2: Schematic representation of the construction of the and
estimators of
on a block of five adjacent
returns.

A few features are worth emphasizing. First, we display the NT
estimators obtained from the first two order statistics along with
the rest, even if they are excluded from the construction of the
estimator. Hence, the third box displays all five returns taken to
the fourth power. An extreme right skew is evident, with values
spanning 0 to 915.1, even if the initial returns are not
particularly scattered. A zero return is, of course, common due to
the discreteness of the price grid. Second, between box 3 and 4 we
apply the relevant scaling factors for the NT estimators, see Panel
B of Table 1. Strikingly, the
large factor (5.74 = (0.1741)) for the
second order statistic produces, by far, the largest realized
estimator in box 4 (466.2). Finally, excluding the NT estimators
originating from the two smallest absolute returns (0, 466.2), we
pick the minimum and median of the remainder and scale these
statistics (78.9 and 163.3) suitably with the scaling factors
provided in Panel D of Table 1 (2.611 =
(0.38303) and 1.214 = (0.82367)) to obtain the local RNT estimators for
of 206.1 given by ,
respectively 198.3 given by . These realizations happen to stem
from the two largest order statistic of the original return block
(5.5 and 4.5), but the low scaling factors for these order
statistics (0.086 = (11.59249) and 0.398 =
(2.51102)) imply that the associated (unbiased)
NT estimators in box 4 are the smallest among the relevant subset.
It reflects the relatively low spread between the three largest
absolute return realizations of 4.0, 4.5, and 5.5. In general,
these procedures tend to moderate the local estimates relative to
estimators which rely more directly on the raw fourth powers in box
3.

Table 1a: Tabulation of Moments of Order Statistics for Standard Gaussian Return Blocks of up to Five Returns - Panel A:
2nd Moments Defining the Inverse Scaling Factors for Corresponding NTV Estimators

Block Size

Z2(1)

Z2(2)

Z2(3)

Z2(4)

Z2(5)

2

µ2(1,2) ≈1.6366198 =π - 2/π

µ2(2,2) ≈0.36338023 =2 + π/π

3

µ2(1,3) ≈0.19279847 =-6+2√3+π/π

µ2(2,3) ≈0.70454374 =6-4√3+π/π

µ2(3,3) ≈ 2.1026578 =1+2√3/π

4

µ2(1,4) ≈0.12070214 =1+4(4√3-9)/3π

µ2(2,4) ≈0.40908747 =12-8√3+π/π

µ2(3,4) ≈1

µ2(4,4) ≈2.4702104 = 1 + 8/ √ 3π

5

µ2(1,5)≈0.083077313

µ2(2,5)≈0.271201456

µ2(3,5)≈0.61591649

µ2(4,5)≈1.2560557

µ2(5,5)≈2.7737491

Table 1b: Tabulation of Moments of Order Statistics for Standard Gaussian Return Blocks of up to Five Returns - Panel B:
4th Moments Defining the Inverse Scaling Factors for Corresponding NTQ Estimators

Block Size

Z4(1)

Z4(2)

Z4(3)

Z4(4)

Z4(5)

2

µ4(1,2)≈0.45352091 =3-8/π

µ4(2,2)≈5.5464791 =3+8/π

3

µ4(1,3)≈0.13874649 =3+26-24√3/√3π

µ4(2,3)≈1.0830697 =72-52√3+9π/3π

µ4(3,3)≈7.7781838 =3+26/√3π

4

µ4(1,4)≈ 0.057664089 =3+4(4(13√3-27)π-9/9π2

µ4(2,4)≈0.38199370 =3+4(9+(36-26√3)π)//3π2

µ4(3,4)≈1.7841458 =3-12/π2

µ4(4,4)≈9.7761964 =3+4(9+26√3π)/9π2

5

µ4(1,5)≈0.028554808

µ4(2,5)≈0.17410122

µ4(3,5)≈0.69383242

µ4(4,5)≈2.5110214

µ4(5,5)≈11.592490

Table 1c: Tabulation of Moments of Order Statistics for Standard Gaussian Return Blocks of up to Five Returns - Panel C:
2nd Moments Defining the Inverse Scaling Factors for Corresponding RNTV Estimators

Block Size

min( Z2(3) / µ2(3,5) ,Z2(4) / µ2(4,5) ,Z2(5) / µ2(5,5) )

med( Z2(3) / µ2(3,5) ,Z2(4) / µ2(4,5) ,Z2(5) / µ2(5,5) )

5

µ2(1,(3,4,5)) ≈0.62084

µ2(2,(3,4,5)) ≈ 0.94544

Table 1d: Tabulation of Moments of Order Statistics for Standard Gaussian Return Blocks of up to Five Returns - Panel D:
4th Moments Defining the Inverse Scaling Factors for Corresponding RNTQ Estimators

Block Size

min( Z4(3) / µ4(3,5) ,Z4(4) / µ4(4,5) ,Z4(5) / µ4(5,5) )

med( Z4(3) / µ4(3,5) ,Z4(4) / µ4(4,5) ,Z4(5) / µ4(5,5) )

5

µ4(1,(3,4,5)) ≈0.38303

µ4(2,(3,4,5)) ≈ 0.82367

We compute the second and fourth moments of order statistics based on blocks of powers of independent standard normals, Zi~N(0, 1), whose inverse represent the scaling factors of the NT and RNT estimators defined in sections 2.4.3 and 3.1. Panel A: Expectation of order statistics of squared normals (NTV estimators). Panel B: Expectation of order statistics of normals raised to the 4th power (NTQ estimators). Panel C: Expectation of quantiles of rescaled squared order statistics of normals (RNTV estimators). Panel D: Expectation of quantiles of rescaled order statistics of normals raised to the 4th power (RNTQ estimators).

4 Robustification Towards Noise and
Errors

In estimating higher order return power variation measures, we
deal with procedures that can be highly sensitive to erroneous
outliers as well as the presence of noise. Hence, we adopt various
techniques that mitigate the impact of such features on the
inference. Our strategy includes standard pre-filtering for obvious
data errors, pre-averaging to reduce the magnitude of the noise in
the returns, and conducting inference on the ratio of versus rather than directly for
. However, most inference techniques
continue to display excessive sensitivity to data irregularities.
Consequently, we supplement the above steps with a novel filtering
method, specifically designed for robust power variation estimators
operating on return blocks. This section reviews the techniques we
employ to enhance the robustness of our inference towards data
errors and noise.

4.1 Eliminating Obvious Errors in the
Tick-by-Tick Data

Any large set of raw transactions data is invariably subject to
recording errors that infuse noise into the high-frequency returns.
Most dramatically, faulty prices create artificial outliers,
causing so-called "bounce-backs" in returns, as there is a
"jump" both when the flawed price first appears and later, often
shortly thereafter, when the price reverts to the correct level.
Hence, the need for effective cleaning procedures has long been
acknowledged. BNHLS (2009) lay out a systematic framework for
dealing with trade data from NYSE-TAQ. In their terminology, we
apply the filters P1-P3 and T1-T4.15 These filters are
arguably mild and uncontroversial and simply aim to eliminate
obvious data errors.

4.2 Pre-Averaging

The assumption that (observed) high-frequency returns embody a
diffusive component is systematically violated at the tick-by-tick
level due to various market microstructure features, including the
finite price grid and the bid-ask spread. As a result, tick-by-tick
price changes are often an order of magnitude larger than what is
consistent with a diffusive characterization. One effective
approach to mitigating the impact of such noise is to apply
pre-averaging, as originally suggested by Podolskij and Vetter
(2009a). This is achieved by transforming the noisy observations on
ultra high-frequency returns into a smaller set of kernel-averaged,
and thus less erratic, "smoothed" returns. In particular, each of
the returns within a block are obtained via
kernel-averaging based on separate, non-overlapping subsets of
tick-by-tick returns. The benefit is a reduction in the impact of
idiosyncratic noise and, especially, distortions induced by
bounce-backs. The drawback is a substantial drop in the underlying
sampling frequency. The latter impacts the choice of the window
width, , as the (diffusive) volatility fluctuates
more widely across longer blocks.

Our implementation of pre-averaging, detailed in Appendix 6,
is based on a relatively conservative choice of sampling frequency.
This has the effect of emphasizing noise robustness over
efficiency. Importantly, it also simplifies the analysis, as the
impact of noise may be largely ignored in the asymptotic theory.
First, the pre-averaging estimator has an asymptotic bias, but if
the (pre-averaged) returns are not sampled at very high
frequencies, the bias is, effectively, negligible. Second, if there
are original high-frequency returns, the
optimal convergence rate for pre-averaged estimators in the
presence of noise is typically . The
associated asymptotic variance reflects both the sampling variance
of the true returns and the noise variance. This result is obtained
if the number of returns per pre-averaging block, , grows at the asymptotic rate , so that the total number of pre-averaged returns
without overlap, , also is of order . This allows the convergence rate - as usual - to
equal the square-root of the number of (pre-averaged) returns,
i.e.,
. But if
is larger, asymptotically rising at the rate for
, the number of
pre-averaged returns grows more slowly,
, implying a convergence rate
of
, e.g.,
for
. At the same time, the noise
will be averaged more aggressively and vanishes asymptotically at a
faster rate. The bottom line is that, by appealing to a slower
asymptotic convergence rate relative to the number of original
high-frequency returns, the (asymptotic) efficiency is lower, but
the asymptotic variance of the pre-averaging estimator becomes
identical to the one for the no-noise case with
returns. However, this equivalence holds only for pre-averaged
return series based on non-overlapping blocks without sub-sampling.
The additional efficiency gain attainable by sub-sampling, as in
Appendix C, is not
identical with and without pre-averaging, differs from one
estimator to another, and generally is not known in closed
form.16 Nonetheless, the efficiency of each
pre-averaged and sub-sampled estimator in the presence of noise is
very close to its efficiency in the absence of noise, as long as
the pre-averaging window size is sufficiently large relative to
sample size. We monitor the latter prediction in the simulations
below to verify that it provides a useful characterization of the
relevant features of the finite sample distribution.

In summary, we appeal to an asymptotic theory guided by a
slightly slower convergence rate than the "optimal"
for pre-averaged estimators. This
enhances robustness to noise while allowing the asymptotic theory
for the no-noise case to be the relevant benchmark. In practice, we
choose a relatively large return block so our procedure is
compatible with the theoretical setting in this regard. This has
the convenient implication that the theory in Sections 2 and 3
provides the appropriate basis for assessing the limiting behavior
of our estimators computed from pre-averaged returns, even if it
ignores the presence of noise. Thus, henceforth, we simply treat
the pre-averaged returns as if they were the original raw returns
and, with slight abuse of notation, we redefine
to denote the relevant number of (pre-averaged) returns, while
accommodating the effect of sub-sampling in the conventional
fashion.

4.3 Filtering via Truncation of Return Functionals

Even for returns based on pre-averaged tick data and sampled at
moderate frequencies, microstructure features and other data
irregularities may induce inhomogeneous and serially correlated
observations that blatantly violate our distributional assumptions.
For quarticity estimation, in particular, it is paramount to
control the impact of this type of data imperfections to achieve a
beneficial trade-off between robustness and efficiency.

This section briefly outlines a general truncation principle for
return functionals that enhances the robustness of integrated power
variation estimators operating on return blocks. It provides an
extension of existing techniques that employ truncation to
alleviate the impact of jumps or data errors. However, the
philosophy and implementation are very different. Existing
procedures truncate returns based on whether a single observation
constitutes a significant outlier under the local Brownian null.
Moreover, the truncation is an essential step in rendering the
estimator robust as it dampens, and asymptotically eliminates, the
distortion induced by price jumps on the estimated power variation.
For this to be effective, the detection of larger jumps must be
reliable, and it is common to apply a threshold for jumps that
correspond to "three sigmas" or a p-value of about 0.3%. As a
result, the procedure generates a non-trivial incidence of type I
errors because diffusive returns based on high-frequency return
data inevitably are subjected to unwarranted truncation.17

In contrast, we develop a filtering procedure that operates
directly on the jump-robust functional and more broadly alleviates
distortions induced by deviations from the null that the block
consists of i.i.d. draws from a normal distribution. In this
scenario, jump robustness is, in principle, already assured by the
choice of an appropriate functional. Hence, the filtering is merely
intended to eliminate truly excessive ex post estimates of
local power variation, driven by functional values incompatible
with the maintained null hypothesis. As such, we rely on an
extremely conservative threshold for truncation, typically with
p-values around or below. This is
sufficiently low that we expect, under the null hypothesis, to
truncate less than a single realization of the return functional
across our entire sample. In practice, the underlying assumptions
are violated and truncation occurs with non-trivial frequency which
helps control the associated distortion in the power variation
estimators.

To introduce this filtering procedure, we recall that
is a functional
providing an unbiased estimator of the local power variation,
under the null hypothesis.
Next, for a sufficiently small , e.g.,
, we let
denote the
th-quantile of the distribution
of a random variable . We then define the
corresponding truncated functional
,

Accordingly, the realized truncated estimator based on
is given by,

where
equals one if the
expression
is true and zero otherwise.
Setting we obtain the usual realized
estimator based on without truncation.
Moreover, if our functional filtering is
equivalent to the usual return filtering at the significance level
.

A feasible version of the filter is developed in Andersen,
Dobrev and Schaumburg (2011). The procedure exploits a local
estimate of volatility based on preceding observations to provide
the appropriate truncation level - exactly as done for the standard
truncation RV estimator - while simulation is performed to obtain
the critical values, taking into account the presence of estimation
error for local volatility.

4.4 Using the Ratio
for Robust
IV Inference

The primary applications of IQ estimation is to draw inference
about IV and to test for jumps under the null hypothesis of no
jumps. For these procedures to perform well, it is essential that
the IQ estimator has good efficiency and finite sample jump
robustness.

Let
,
be suitable jump-robust
estimators of IV and IQ. A natural approach for drawing inference
about IV follows directly from its limiting distribution,

where the "efficiency" factor, , depends on the
specific choice of estimator.

Letting RV denote the realized volatility estimator, which is
the efficient estimator of IV under the null, the natural Hausman
test statistic for the presence of jumps, see BNS (2004) and Huang
and Tauchen (2005), is given by

An asymptotically equivalent set of test statistics with better
finite sample properties, proposed by BNS (2002), can be derived by
applying the delta method to the log-transform of the volatility
measures. This has the benefit that IQ enters only in terms of the
ratio
which, as also
demonstrated in our empirical investigation below, has a
stabilizing effect on the variance of log
.

The corresponding Hausman test statistic for the presence of jumps
is

While the literature has documented superior performance of this
ratio for jump-robust inference in a frictionless setting, it is
evident that the ratio also will impact the way market
microstructure noise affects the inference. In Appendix B, we
provide an illustration based on computations involving the
non-robust versions of the IQ and IV estimators. The findings point
towards favorable properties of the ratio statistic relative to the
raw statistic along this dimension as well. The intuition is as
before: the realized and
statistics tend to be impacted by noise in similar ways so the
ratio provides a partial cancelation of errors. The issue is
further pursued within the simulation set-up entertained in the
following section.

5 Finite Sample Simulation Evidence

We design a series of Monte Carlo experiments, each focusing on
a distinct feature of the data generating process that may affect
the finite sample behavior of the estimators. The emphasis is on
the qualitative impact of each feature in isolation. In reality,
multiple features interact, creating complex patterns in
tick-by-tick data. The joint presence of various factors, partially
reinforcing or counteracting each other, render it difficult to
infer the significance of individual features. Hence, our
simulation design is not intended to replicate the empirical
results in all dimensions, but to assist in identifying the
features that create differential patterns in the results obtained
from alternative estimators.

5.1 Estimators

We adopt the novel filtering procedure, based on (mildly)
truncating the local power variation functional, for all estimators
except those already truncating individual returns more
aggressively, i.e., the truncation power variation estimators. For
the latter, the impact of an additional layer of mild truncation is
negligible.18 For the truncation estimators, we
follow the approach advocated by CPR, as they document it improves
on existing implementations. Overall, we consider the following IV
and IQ estimators,

Multi-Power Variation: We denote
MPV(m,2) by MPVm and
MPV(m,4) by MPQm. We include
MPV3 and MPV5, as well as
MPQ3 and MPQ5 in our analysis.
This type of estimators was introduced by BNS (2004);

Neighborhood Truncation: We consider
MedRV = NT(2,3,2) and the
corresponding quarticity estimator MedRQ =
NT(2,3,4). They are developed in ADS (2012)
and in this paper;

Robust Neighborhood Truncation: We use = RNT
, =
RNT
, =
RNT
, and
= RNT
. They
are developed in this paper.

In addition, when investigating IV estimators, we include the
standard RV estimator, serving as a non jump-robust benchmark,
along with the QRV estimator of COP (2010). We omit an IQ
counterpart of QRV from our analysis because we find the block size
of 20 or more returns, necessary in order to establish the
quantiles, to be prohibitively large for reliable inference on
actual data subject to irregular sampling and pronounced intraday
variation in volatility. Moreover, as discussed previously, one may
interpret our NT estimators as modified QRV estimators based on the
absolute returns over very small blocks.

Finally, taking into account the need to apply suitable
noise-reduction technique when conducting inference in practice, we
focus our Monte Carlo analysis exclusively on the pre-averaged
implementation of all estimators, as defined in detail in Appendix C,
including the efficiency gain from sub-sampling. The use of
pre-averaging necessitates a somewhat conservative choice of
sampling frequency. We report results based on 30, 120 and 600
second return observations.19

5.2 Simulation Results

We largely follow the comprehensive simulation design in ADS
(2012) adopted for comparing IV estimators. We calibrate the
unconditional daily IV to , or roughly
20% per year, across all scenarios. For each scenario we simulate
250,000 trading days, corresponding to about 1,000 years, from 9:30
am to 4:00 pm with new prices arriving every 3 seconds on average,
so we have 7,800 distinct prices each day.

We consider three major departures from the Gaussian benchmark:
(i) microstructure noise: bid-ask bounce,
recording errors, irregular trade intervals, and price
decimalization (discreteness); (ii) time-varying
volatility: stochastic and deterministic (diurnal)
variation in volatility along with volatility jumps; (iii)
jumps in returns: one or multiple intraday price jumps.
Each scenario is briefly described below, with additional details
available in ADS (2012). We focus on estimators of , , and the ratio
, with the latter computed using
the same type of estimator for the numerator and
denominator.20 For brevity, we often - including in
the tables - refer to the ratio as estimated by, say, , when it is estimated by
.

5.2.1 The Brownian Motion Benchmark

This is our baseline scenario with sampling on an equispaced
time grid. It is an ideal setting in which we expect the finite
sample performance of all estimators to closely mimic the
underlying asymptotic theory. Panel A of Tables 2 and 3 verify that most
of the estimators are unbiased for IV and IQ. The exception is the
minor downward bias in TRV, TBV, TRQ and TBQ. This stems from noise
in the truncation and bias-correction procedures applied in
constructing these estimators. They truncate individual returns at
three (estimated) standard deviations, so a scaling is needed to
mitigate the impact of erroneous truncation of diffusive returns
and this introduces some estimation error, even under the BM null.
All other estimators rely on the very conservative truncation level
(p-value of ) associated with our functional
filtering, described in Section 4.3. As
intended, the impact of this filter is negligible under the null
hypothesis, so the estimators remain unbiased.

In terms of efficiency, the ordering of the jump-robust
estimators is as prescribed by theory, with TRV superior in terms
of RMSE for IV (matching RV), followed by , QRV, , and TBV. For IQ,
TRQ is best, followed very closely by , and then , and TBQ at the
higher frequencies.21 In summary, the RNT estimators
perform well and, in particular, come close to matching the
efficiency of TRQ for IQ. It suggests that this type of estimator
can enhance robustness to noise and jumps without any significant
loss of efficiency in general.

Finally, turning to the estimates for
in Table 4, Panel A, we
notice a small downward bias at the lowest sampling frequency in
almost all cases. This is caused by a finite sample Jensen
effect.22 More remarkably, in terms of
efficiency, the TRQ estimator now performs relatively poorly.
Instead, and outperform the alternatives by a substantial margin.
The dramatic shift in relative efficiency reflects the fact that
cancelation of outlier terms in the numerator and denominator is
particularly effective under the robust neighborhood truncation
principle where the largest return realizations are prevented from
exerting any significant impact. Likewise, the MPQ5 estimator
performs quite well due to the effective dampening of outliers. We
conclude that, even under ideal circumstances, the RNT estimators
provide an attractive alternative to existing procedures,
especially for inference and jump tests based on the
ratio.

5.2.2 Jumps in Returns

To assess finite sample jump-robustness, we augment the BM model
with return jumps of the Poisson-Gaussian type that are independent
of volatility and account for 20% of the daily QV (25% of IV). We
focus on two cases, one with a single jump per day, the "BM + 1
Jump" scenario, and one with four jumps per day, the "BM + 4
Jumps" scenario, but the overall jump contribution to the daily
variance is identical for the two cases.

Panels B and C in Table 2 show that TRV, TBV, QRV, RMinRV and RMedRV provide the best robust IV inference
in terms of RMSE. Moreover, the relative performance within this
group shifts as we move from a single to four jumps with, in
particular, RMinRV and TBV improving their standing as the jump intensity
increases. This tendency is even more pronounced for IQ estimation,
where panels B and C in Table 3 reveal that RMinRQ is the best performer in both jump scenarios at the
2-minute frequency.

Finally, and very strikingly, panels B and C of Table 4 show that the
pairing of RMinRQ and RMinRV dominate all other estimators by a significant
margin in terms of estimating the ratio
, which governs the precision of
log(IV). Thus, from the perspective of finite sample
jump-robustness, the RNT estimators seem to offer attractive
efficiency improvements, especially for the estimation of IQ and
.

Juxtaposing Panel B or C in Tables 3 and 4, we also note
that the distortions induced by jumps are much less pronounced for
the ratio statistic than for IQ. Again, the partial cancelation of
the (upward) bias in the numerator and the denominator is
operative. Thus, ratio-based inference is likely preferable
regardless of the choice of estimator. In summary, estimating the
ratio statistic using RMinRQ emerges as a natural part of practical
jump-robust inference for IV or testing for jumps. Below, we
explore whether this estimator is robust to other common
"irregularities" in high-frequency return data as well.

5.2.3 Time-Varying Volatility

Pronounced intraday variation as well as seemingly abrupt
changes (jumps) in spot volatility are prevalent in high-frequency
returns. This poses a challenge for power variation estimation, as
jump-robust estimators may not be able to distinguish sharply
between rapidly shifting volatility and return jumps. For example,
in the context of IV estimation, ADS (2012) document sizeable
finite sample distortion in IV estimators when the intraday
volatility is stochastic and subject to a diurnal U-shaped pattern.
This section extends this analysis and draws broader conclusions
for estimation of IQ and
by exploring two distinct
scenarios that violate the (locally) constant volatility
assumption.

Our first scenario, "SV-U," is a modification of the
corresponding design in ADS (2012). The diurnal pattern is
calibrated to the average volatility of tick-time sampled trades of
the stocks analyzed in the next section. In particular, we simulate
a two-factor affine stochastic volatility model and superimpose an
asymmetric diurnal pattern (Hasbrouck, 1999) for which the variance
at the open is more than four times the midday and end-of-day
variance.

The second scenario, "BM + 1 Volatility Jump," involves a
six-fold spike in the intraday variance at a random point in time,
uniformly distributed across the trading day. Volatility is
constant before the jump, and then constant at the new higher level
following the jump. In this way, the scenario approximates the
effect of sudden bursts in market activity that have inspired the
development of alternative volatility jump specifications.

The striking similarity between Panels D and E of Tables 2-4 indicates that
these two distinct forms of time variation in volatility have a
qualitatively similar impact on the estimators in terms of finite
sample bias and RMSE. Effectively, both scenarios render
neighboring returns inhomogeneous, resulting in a downward bias due
to scaling factors that are incorrectly sized as well as
inappropriate truncation of diffusive returns that are
misclassified as jumps due to the fluctuating level of the return
variance. The less "local" estimators are more exposed to such
heterogeneity. This explains the ordering of the biases of the IV
and IQ estimators in Panels D and E of Tables 2 and 3, with estimators
relying on block size one to three being the least biased, those
based on blocks of four or five returns being slightly more biased,
and finally the estimators relying on substantially larger block
sizes (such as 20 for QRV in Table 2)
being most biased.

In summary, Panels D and E of Table 3 provide evidence against the use of sparser sampling frequencies,
such as ten minutes or lower, for IQ estimation. This runs counter
to suggestions in the literature, indicating that biases in IQ
estimation may be alleviated through sparse sampling. We find, in
contrast, that the bias is much lower, and quite tolerable, at the
two minute frequency, regardless of the block size of the IQ
estimator.

Most importantly, Panels D and E in Table 4, confirm that
the biases for the ratio
are less pronounced and more
uniform across the full range of estimators, as may be expected
given that we obtain partial cancelation of the downward biases
which affect both the numerator and denominator. Moreover, the
ratio estimator based on and again performs best from an efficiency
standpoint in spite of the block size of five. For comparison,
increasing the block size of the MPQ and MPV estimators from three
to five enlarges the RMSE for
in the "BM + 1 Volatility
Jump" scenario. Consequently, the superior efficiency in
estimating
stems from the design of the
robust neighborhood truncation principle rather than from the
increased block size.

We conclude that inference based on the ratio
appears to be attractive also
under time-varying volatility. Moreover, the estimator provides
quite compelling performance in this setting as well for the higher
sampling frequencies.

5.2.4 Microstructure Noise

There is a tradeoff in the choice of sampling frequency with
jump robustness improving and resiliency to microstructure noise
deteriorating as the return interval shrinks. We now explore the
effectiveness in dealing with the adverse impact of various noise
features by applying pre-averaging, as detailed in Appendix C,
and sampling at moderate frequencies. In fact, as outlined in
Section 4.2,
our asymptotic theory is developed in a noise-free setting, so we
seek to determine whether this provides a suitable approximation
for practical inference.

We consider four separate market imperfections. First, in our
"BM + IID Noise" scenario, Panel F, Tables 2-4, we simulate
Gaussian i.i.d. noise with a noise-to-signal ratio of
, in line with what is typical
for trade data on individual stocks. Second, we consider a "BM + 1
Bounceback" scenario, Panel G, Tables 2-4, in which
(isolated) errors in the recorded price induce so-called
"bounce-backs" in returns, i.e., two large adjacent jumps of
opposite sign due to immediate price reversals. We calibrate the
magnitude of the bounce-back to match 20% of the daily QV (25% of
IV). The third source of noise is irregular sampling, and the
associated results are captured by our "BM + Sparcity" scenario,
Panel H, Tables 2-4. It is generated
via random arrivals of the 7,800 distinct quotes by sampling
without replacement from the numbers in the range of 1 through
23,400. While not necessarily realistic, this model is helpful in
exploring the potential distortion of the estimators when applied
on non-homogeneously sampled returns, effectively inducing spurious
variations in their volatility. Finally, in our "BM + Discrete
Pricing" scenario, Panel I, Tables 2-4, we mimic price
decimalization by rounding all intraday prices to the nearest cent
with a starting price of $50. Price discreteness is a major reason
for the presence of multiple zero returns in high-frequency
samples, leading to pronounced downward biases of many jump-robust
estimators.23

First, the reported relative biases in Panels F, G, H, and I of
Tables 2-4 reveal that,
irrespective of the noise scenario, it is necessary to avoid
sampling at the highest frequency, i.e., 30 seconds, to obtain
reasonably unbiased estimates of IV and IQ, while the ratio
is unbiased for all scenarios
except the "sparsity" setting. However, once we reach 2 minutes,
all the relevant quantities are estimated without bias, except for
a minor bias for IQ in the sparsity scenario.

Second, for the 2-minute frequency, the MSE is nearly identical
for the Brownian motion case and the various noise scenarios,
highlighting the efficacy of the pre-averaging, filtering and
subsampling procedures. In particular, for the ratio statistic, the
MSE is literally identical across all scenarios and estimators at
the 2-minute frequency except for the sparse sampling case. That
is, noise has no discernible impact on the asymptotic errors of the
estimators at this moderate frequency, apart from minor distortions
arising from inhomogeneous sampling of the returns. In the
empirical work below, we mitigate this effect by sampling in tick
time which renders the return variability more uniform across
observations.

Third, across all noise scenarios, the relative bias and MSE for
the ratio
are dramatically lower than for
the IQ estimators. Thus, the cancelation of outliers in the
numerator and denominator helps robustify the ratio statistic in
the presence of noise.

Finally, we note that the pre-averaging is extremely effective
for the "BM + 1 Bounceback" scenario. This is due to the near
perfect cancelation of adjacent jumps of opposite sign when
constructing the individual pre-averaged returns. In what follows,
we rely on pre-averaging as implemented in our simulation
experiments to suppress the impact of noise also in our empirical
illustrations on real market data.

6 An Illustration for the Dow Jones 30 Stocks

Since the "true" values of IV and IQ are latent, there is no
simple way to directly compare the performance of alternative
estimators. Moreover, for IQ in particular, there is a great deal
of uncertainty regarding the actual precision of existing
estimation procedures. The preceding analysis has focused on
bringing out the features that render estimation inaccurate as well
as non-robust, and then developing new approaches that should
improve the inference. We now seek to establish whether the issues
we have identified actually do pose a challenge for practical
estimation and if the suggestions and procedures we have proposed
appear helpful.

Consequently, this section explores properties of competing
estimators of , , and
for the Dow Jones 30 stocks
using tick time sampling of NYSE/TAQ trade data.24 We
split our sample period into a low volatility period, January 2005
- May 2007, and a high volatility period, June 2007 - July 2009.
This serves as a robustness check against different noise-to-signal
ratios and liquidity levels in the two periods.25 The focus is on
the estimates for and
, while the
results provide a benchmark for assessing the RNT estimators as
well as the impact of our new filtering scheme relative to prior
findings.

6.1 Truncation of Return
Functionals

As documented in Section 5.2.4, power variation
estimators can be quite sensitive to deviations from the local
diffusive null, arising from microstructure features or recording
errors. Common data filtering procedures may eliminate extreme
outliers, but they are not sufficient to ensure sensible IQ
estimates in practice for many candidate estimators of
interest.

Figure: 3: IQ Signature Plots

In Panel A, the estimation is performed as
described in the paper and, in particular, all estimators, except
TRQ and TBQ, are subject to the functional filtering procedure
detailed in Section 4.3. In contrast,
Panel B depicts the estimators without functional filtering. TRQ
and TBQ are identical in the two panels, as they are based solely
on truncation of individual returns. The average is across all
stocks in the DJ30 index during January 2005-May 2007 and all
estimates are pre-averaged and sub-sampled based on tick-time
sampling.

Panel A of Figure 3 depicts
signature plots for a group of IQ estimators, obtained by averaging
the daily IQ estimates across the entire sample period and all of
the thirty stocks. By construction, the figure speaks to mean and
bias effects rather than efficiency. It shows that the estimators
generally are quite similar although it also reveals some
significant variation. First, at the highest frequencies a few
estimators, especially MPQ5, appear severely downward biased.
Hence, the noise-reduction associated with pre-averaging,
sub-sampling and filtering has been successful in stemming the
upward bias of the raw estimators. The remaining effects are
consistent with the impact of irregular or sparse sampling of the
returns at the highest frequencies, see Table 3,
Panel H. The other striking feature is the slow decline in the
plots as we move towards lower frequencies. The most likely
explanation is the impact of time-varying volatility within the
sampling interval, as indicated by the results in Table 3,
Panels D and E. We pursue this issue in detail in the following
section. Nonetheless, it is noteworthy that the signature plots are
quite flat and largely coincide for the various estimators in the
range of 90-150 seconds.

By contrast, Panel B of Figure 3 offers widely
diverging results. Here, none of the estimators are subject to
functional filtering, so only TRQ and TBQ provide uniform
truncation of stark outliers. The consequence is apparent. Apart
from the extreme dampening achieved by MPQ5, the remaining
estimators are wildly upward biased at the higher frequencies. This
speaks to the lack of robustness of IQ estimators that do not
exploit direct truncation of individual returns. The problem of
excessive variability, or noise, in IQ estimates has been noted
sporadically in the empirical literature and it has motivated some
authors to rely on low, and relatively inefficient but less error
prone, frequencies for IQ estimation, e.g., BNS (2004a) and Bandi
and Russell (2008). Similarly, Jiang and Oomen (2008) uses the
squared IV as a simple approximation to IQ, thereby accepting a
significant bias in exchange for variance reduction of the IQ
estimator.

Thus, it is very encouraging that the functional filtering
regularizes the IQ estimators. The nominal size of our filter is
, so only gross violations of the
Gaussian null is flagged. However, obviously, the null hypothesis
is not satisfied for actual high-frequency data, so the truncation
frequency is substantially larger in practice. For our equity data,
the fraction of observations filtered ranges from to for the 60-180 second range,
depending on the frequency and sample period. Overall, more than
70% of the stock days are untouched by our functional truncation.
In contrast, TBQ truncates at least one observation on 99.9% of the
days.26 Overall, the evidence is compatible
with our objective, namely that the functional filtering should
control major data irregularities while avoiding excessively
intrusive, and potentially distorting, truncation of the underlying
returns.

6.2 The Intraday Volatility
Pattern

One benefit of tick-time sampling of transactions data is that
it tends to mitigate the intra-day U-shape pattern in volatility.
Figure 4 demonstrates
that the tick-time sampling succeeds in straightening the
volatility pattern across the main part of the trading day, but
there is little impact on the elevated level of volatility in the
first 60-90 minutes of trading. Our Monte Carlo experiments found
such intraday volatility variation to be a potent source of
systematic biases in power variation estimates, with the ratio
statistic
being less sensitive to such
distortions than the raw and
measures.

Figure 4: Diurnal volatility pattern for intraday trade data across the DJ30 stocks
between January 1, 2005 and May 31, 2007.

We plot the diurnal U-shape variance factors across stock-days based on local estimates
of
in one minute buckets using tick
time (Panel A) or calendar time (Panel B) sampling. On each
stock-day, the factor in each one-minute bucket is computed by
normalizing by the average of the 390 variance estimates on that
day. The average variance factor is then computed by averaging
across all stock-days

Data for Figure 4, Panel A: Tick Time

Intraday Bin (min)

MPV3

MPV5

MedRV

TRV

TBV

RMinRV

1

6.064729982

6.022776011

5.341480764

6.164035434

5.777039144

5.004557031

2

4.613565484

4.882487692

4.169216556

4.454525454

4.403919085

4.142636973

3

4.084005134

4.337520533

3.673024203

3.890645523

3.891508601

3.714971954

4

3.780395522

3.937862803

3.403077789

3.517630798

3.570706647

3.426637816

5

3.506969572

3.677154387

3.221589471

3.364749558

3.374114756

3.233007102

6

3.267677058

3.499849224

3.012623121

3.163501366

3.173633628

3.065716301

7

3.093526251

3.324252024

2.828995683

2.993073099

2.963303811

2.89460483

8

2.959586019

3.235256365

2.7008724

2.798166335

2.840572381

2.777552384

9

2.843499457

3.043338114

2.616089934

2.701745173

2.745091074

2.674407739

10

2.753303917

2.921953112

2.541200946

2.642641651

2.63482782

2.593835087

11

2.607607721

2.830135698

2.40635473

2.526060076

2.550188722

2.495426477

12

2.527600851

2.692385034

2.362544699

2.448782731

2.446077956

2.412963322

13

2.44955126

2.572271387

2.27571698

2.333683131

2.358744897

2.326788387

14

2.384465484

2.523862332

2.214671271

2.314846403

2.336624934

2.260742323

15

2.284930017

2.41514381

2.163225949

2.269376317

2.252272265

2.197086703

16

2.219770939

2.361207251

2.062121631

2.153187051

2.143612188

2.137509463

17

2.154926618

2.323151138

2.045166434

2.134803291

2.150363973

2.12183224

18

2.12863631

2.281723383

1.979768603

2.098052291

2.056446669

2.075473317

19

2.08747205

2.210894078

1.953128198

1.987996162

2.012154711

2.033085551

20

2.077482269

2.190707709

1.96420105

1.994568794

2.028905116

2.009589393

21

2.041917845

2.184298198

1.924936855

1.988238035

2.004059556

1.979040903

22

2.018129197

2.141443598

1.910358598

1.965989605

1.967640645

1.955905014

23

1.993637509

2.07286322

1.880109851

1.901085625

1.941615837

1.920921853

24

1.948927115

2.028504283

1.828125057

1.864218195

1.90552374

1.884011271

25

1.904632673

2.016084587

1.807464668

1.874103006

1.896089194

1.852760088

26

1.849826117

1.968689969

1.761693733

1.831899175

1.828594129

1.834099072

27

1.829757518

1.958663154

1.736175711

1.780875062

1.78428099

1.800323376

28

1.82491881

1.930477232

1.724054015

1.769055121

1.776376324

1.78789884

29

1.827960611

1.954213682

1.716579262

1.760321616

1.763630811

1.777254617

30

1.8209282

1.968023005

1.709991848

1.721954478

1.76720133

1.779292681

31

1.804548083

1.962643952

1.714283934

1.727049299

1.761461176

1.778709346

32

1.801821742

1.915146493

1.702068939

1.753121313

1.748199099

1.765887984

33

1.795758994

1.876877507

1.702644778

1.662108498

1.727391864

1.757239536

34

1.79778912

1.869808156

1.709012691

1.729752976

1.750542213

1.741131745

35

1.752540222

1.855105318

1.70199856

1.716478522

1.736422154

1.726154209

36

1.726994938

1.840855521

1.643513403

1.69513222

1.690101947

1.714808957

37

1.694095711

1.856223734

1.631958524

1.681018864

1.672103811

1.710465175

38

1.721163978

1.836336679

1.629358835

1.635864252

1.636273486

1.688399664

39

1.72436076

1.788772264

1.631005903

1.635711574

1.676229061

1.683861655

40

1.705467333

1.809832904

1.623944847

1.66134682

1.680092318

1.679422869

41

1.660650218

1.791071363

1.578582792

1.644143242

1.638521649

1.653240657

42

1.683891311

1.769933551

1.597288334

1.589464952

1.597020722

1.626355896

43

1.680157348

1.7531881

1.594193687

1.579436933

1.640393218

1.606804628

44

1.654379468

1.723502436

1.561082818

1.619203232

1.628674094

1.587814124

45

1.593169138

1.664343538

1.533317966

1.549926713

1.564845776

1.558810123

46

1.565620551

1.634144283

1.516620091

1.523269948

1.538248256

1.537039869

47

1.524130087

1.604629903

1.488526042

1.499681391

1.502518185

1.516667073

48

1.522978048

1.599818722

1.480837745

1.486209955

1.495305177

1.497418333

49

1.513444478

1.572962598

1.442312163

1.467533618

1.486847493

1.468895612

50

1.483929856

1.551094663

1.418349876

1.454030072

1.458162533

1.456255108

51

1.474694267

1.537144598

1.397262004

1.419461809

1.447853838

1.442377522

52

1.449493739

1.496033654

1.377755252

1.402583829

1.409665643

1.427677699

53

1.425656454

1.481856932

1.389670032

1.378111143

1.411060539

1.399687103

54

1.420637605

1.465760587

1.365262103

1.382436403

1.394402854

1.386084074

55

1.39778158

1.438308217

1.351397355

1.376732392

1.385432775

1.367253583

56

1.375205613

1.400733665

1.315731151

1.355752551

1.355041385

1.33726062

57

1.330809327

1.374923611

1.302294972

1.312109051

1.330563902

1.316620474

58

1.321031456

1.377600697

1.276626229

1.302223255

1.307406951

1.313068879

59

1.299486101

1.351832209

1.264951513

1.300893959

1.286863667

1.308990925

60

1.307347206

1.343865335

1.268215957

1.260610089

1.277584017

1.298476334

61

1.318354422

1.361837511

1.269006212

1.27571242

1.299444804

1.299428313

62

1.288785753

1.355890605

1.270342929

1.274801086

1.286946821

1.288077696

63

1.293812028

1.347095813

1.271605672

1.252751242

1.269503556

1.284965186

64

1.298657247

1.349930462

1.260809332

1.261070125

1.276610497

1.292905736

65

1.293666032

1.331554438

1.25821166

1.261697157

1.27920466

1.277354777

66

1.290083641

1.301687497

1.246686877

1.245738999

1.259321623

1.27038105

67

1.25468691

1.294865077

1.244572873

1.247926202

1.25945725

1.256375147

68

1.26577821

1.305356313

1.232668255

1.236556587

1.236907937

1.259125339

69

1.251988057

1.276933223

1.224568964

1.208649929

1.237079825

1.237755853

70

1.259512277

1.280212295

1.226165306

1.218090515

1.235300782

1.226836822

71

1.229897737

1.247469223

1.210250172

1.223270051

1.246310079

1.222111696

72

1.221010848

1.241222012

1.190503769

1.227629561

1.213757218

1.202002821

73

1.185878762

1.217340485

1.161583531

1.160935262

1.17173275

1.202463482

74

1.204479608

1.213876044

1.188381427

1.178127296

1.190187185

1.209053522

75

1.198479256

1.205152157

1.194933179

1.172162943

1.199233582

1.205409648

76

1.192342893

1.189468461

1.194437691

1.186185689

1.189668885

1.19153043

77

1.185546332

1.19194341

1.160545584

1.177928366

1.189120285

1.187227057

78

1.171534703

1.19143832

1.168068298

1.169158836

1.171684011

1.179937998

79

1.171392307

1.187703893

1.144432968

1.140946143

1.154657707

1.159661812

80

1.157895167

1.175767035

1.125309332

1.125993087

1.140843741

1.147873068

81

1.155529186

1.161841365

1.150080206

1.158953531

1.148227195

1.144739957

82

1.142085436

1.153346688

1.129121639

1.121672526

1.130633208

1.138589997

83

1.143299601

1.168033265

1.129155109

1.135443789

1.140109003

1.129968589

84

1.118995223

1.144210787

1.0997482

1.108420975

1.11408785

1.114175711

85

1.123980789

1.149215482

1.102230807

1.109245344

1.120863574

1.106935542

86

1.112133561

1.134469474

1.099104823

1.083770288

1.097256208

1.107082411

87

1.119038941

1.137165223

1.100659058

1.086746639

1.116358466

1.09897768

88

1.09892889

1.117556787

1.086778373

1.084272471

1.093938899

1.093015162

89

1.099424907

1.085823016

1.078557306

1.062903217

1.080136096

1.087617609

90

1.080745505

1.071731821

1.073932646

1.068679712

1.084140203

1.072073304

91

1.054665259

1.075511854

1.059165357

1.06859905

1.069260244

1.059772617

92

1.045655426

1.048687165

1.049740324

1.05256547

1.047980326

1.053186407

93

1.039679399

1.039578387

1.039864115

1.019436545

1.035901017

1.03844283

94

1.04827344

1.041724905

1.048854607

1.031448906

1.049629749

1.039747391

95

1.026012176

1.006883455

1.031169757

1.013641328

1.03425028

1.020822009

96

1.011951973

1.0027134

1.020311474

1.023801037

1.019975637

1.021228394

97

0.9967061

0.981609944

1.009224404

1.003939262

1.008563811

1.005281682

98

0.993736328

0.985098088

1.000389978

1.003754132

1.006866834

0.9958858

99

0.971389736

0.962382956

0.978318433

0.989256488

0.982428382

0.988425397

100

0.968371996

0.949000372

0.974232537

0.968263118

0.974366209

0.976644143

101

0.953848606

0.946874781

0.968067979

0.972655564

0.969487847

0.978709256

102

0.957184032

0.945686181

0.97278091

0.959370092

0.960072564

0.970273096

103

0.956515717

0.947521893

0.968337996

0.961392298

0.960478987

0.967900011

104

0.951682285

0.944712877

0.96050819

0.957015126

0.966223412

0.957089644

105

0.943032276

0.939303924

0.952551213

0.965105613

0.951822488

0.957762808

106

0.936691126

0.925204159

0.94534084

0.937622548

0.933321244

0.952604519

107

0.950569044

0.942597283

0.955318297

0.942161629

0.948424628

0.95702096

108

0.947249606

0.941338958

0.946728019

0.939271888

0.949692738

0.949404478

109

0.940845251

0.923357094

0.940369439

0.937103432

0.938489657

0.947005163

110

0.931787438

0.912509282

0.947118484

0.938400888

0.946730917

0.937194578

111

0.919546132

0.907975835

0.932147092

0.938762625

0.930217837

0.933039272

112

0.915751057

0.907403985

0.923828675

0.922596305

0.920179574

0.927793983

113

0.900918837

0.908106074

0.924045875

0.916468631

0.920294033

0.929196967

114

0.908368862

0.905297304

0.92863022

0.920312278

0.915984867

0.922235378

115

0.911626769

0.888801568

0.927236323

0.913708117

0.912590259

0.91293104

116

0.905056274

0.894860591

0.92736248

0.918776142

0.914643811

0.917057282

117

0.892888437

0.891278446

0.919487338

0.916132405

0.908235461

0.907461917

118

0.896021885

0.875275499

0.911604378

0.889454058

0.891751049

0.895213474

119

0.884495764

0.878596989

0.901594507

0.892556748

0.904905186

0.899203312

120

0.88152932

0.859240337

0.906352479

0.91724525

0.888595627

0.902006957

121

0.893545457

0.870482303

0.902529343

0.875673143

0.882901964

0.898888702

122

0.876513005

0.875935268

0.893604843

0.89196468

0.899328114

0.891797054

123

0.890489284

0.883262093

0.897340468

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0.815257338

0.81416603

0.808662462

0.805116384

286

0.789859669

0.759489556

0.812047221

0.79281351

0.796712414

0.800141057

287

0.779450186

0.751598245

0.813516501

0.794882726

0.801857289

0.802910157

288

0.781148898

0.747712062

0.810888313

0.804078399

0.793851359

0.7988179

289

0.771726718

0.739393763

0.804640696

0.781748199

0.791482834

0.799751855

290

0.778654719

0.758159005

0.802582696

0.811848915

0.795332962

0.800713718

291

0.773971597

0.751542701

0.811701441

0.790198478

0.785181406

0.80008535

292

0.775205875

0.749549075

0.806566949

0.787143025

0.787891507

0.796858168

293

0.779235339

0.762298138

0.801025983

0.797160557

0.786271258

0.798259333

294

0.780382149

0.763353874

0.807765104

0.78856793

0.790384109

0.793011907

295

0.788702679

0.749644685

0.801829166

0.783023548

0.793392851

0.785257131

296

0.773447576

0.736000596

0.805719167

0.790260475

0.792562556

0.776239165

297

0.757427408

0.726847942

0.78774369

0.791239452

0.780868559

0.774533379

298

0.752770215

0.727388822

0.779089801

0.77123491

0.766098849

0.76993422

299

0.755373541

0.714387611

0.788077459

0.76144255

0.765328802

0.765094934

300

0.750852751

0.716427817

0.782974357

0.769173011

0.771744196

0.764347426

301

0.73663583

0.703709265

0.770522683

0.777311671

0.767748622

0.760614291

302

0.733286672

0.700397075

0.769771405

0.762975018

0.747608051

0.757129138

303

0.736972894

0.700431569

0.770615115

0.750613854

0.748147673

0.751243821

304

0.732496775

0.702815923

0.770092554

0.758203133

0.754524012

0.752348045

305

0.742209544

0.713462867

0.770732598

0.764164605

0.758271356

0.757168889

306

0.745178763

0.723072625

0.771069468

0.762759125

0.762983209

0.759691284

307

0.747600714

0.715687756

0.769503065

0.758979551

0.757251979

0.75578832

308

0.743426016

0.704320297

0.776822437

0.751589198

0.75092229

0.75295013

309

0.735026664

0.716833964

0.763416214

0.757606558

0.76210203

0.757284136

310

0.729492366

0.702263528

0.754733393

0.75854404

0.74189654

0.756186358

311

0.727673529

0.706782494

0.763919979

0.746859966

0.739781831

0.754262049

312

0.733999191

0.692251549

0.769241893

0.751780703

0.747974985

0.745273249

313

0.734431956

0.694804457

0.760185154

0.756030043

0.753881792

0.744085469

314

0.718937488

0.681326531

0.755230057

0.756020248

0.746096671

0.739202279

315

0.711377042

0.682980421

0.745768977

0.741823792

0.729256427

0.737653278

316

0.708600021

0.666225654

0.740633241

0.733416619

0.726539009

0.741044208

317

0.713890095

0.67509292

0.750293705

0.742243807

0.72866359

0.742962035

318

0.707914591

0.692202019

0.750801902

0.733710547

0.735411692

0.742018148

319

0.72602678

0.696384085

0.756757273

0.749797295

0.737315146

0.746270804

320

0.731584724

0.695766949

0.756316224

0.736924182

0.739837738

0.741476998

321

0.725333697

0.688437398

0.752761027

0.739150852

0.739416919

0.737378289

322

0.724601328

0.688773091

0.760114412

0.739857744

0.738796267

0.730894569

323

0.71830528

0.685551357

0.752531759

0.743969395

0.737995048

0.725605526

324

0.70764498

0.661037612

0.737524276

0.728665429

0.726755242

0.720485825

325

0.693315558

0.646385087

0.737161395

0.722623908

0.717136883

0.718282459

326

0.686950183

0.632998402

0.729216017

0.723749686

0.713021626

0.71105237

327

0.687692331

0.643141019

0.718883532

0.714159927

0.704894052

0.708313421

328

0.674204057

0.637270374

0.722584411

0.711699237

0.702880561

0.710366089

329

0.677192307

0.637233378

0.719697652

0.704959247

0.693846469

0.703254428

330

0.694241973

0.651248934

0.719146844

0.710442929

0.711574414

0.710664787

331

0.683860138

0.653436584

0.72212674

0.711326056

0.710793945

0.709535865

332

0.676475492

0.646744455

0.715667432

0.711634524

0.699117746

0.70746114

333

0.683210811

0.652560497

0.714438834

0.701808676

0.698034219

0.708759391

334

0.679433196

0.657035126

0.713623675

0.712144285

0.701888832

0.706017945

335

0.691025694

0.647195982

0.720133612

0.708406277

0.707897353

0.699173434

336

0.688011263

0.641308106

0.710598908

0.709685332

0.701686015

0.698211255

337

0.667218441

0.639751644

0.704043312

0.713141299

0.697859775

0.691744725

338

0.671677957

0.638411703

0.713613909

0.713632105

0.68826696

0.696798586

339

0.664495282

0.635886296

0.699303608

0.690362547

0.688106131

0.692869225

340

0.681287044

0.631543896

0.709454402

0.704870758

0.694158017

0.688776684

341

0.66809626

0.630415428

0.704155658

0.686293664

0.693986067

0.684764629

342

0.661736156

0.618431295

0.700167835

0.708544382

0.690738333

0.685337585

343

0.655928572

0.61889661

0.694733234

0.693752763

0.674432954

0.680464521

344

0.652094712

0.612851941

0.686881258

0.676925912

0.668188552

0.676936853

345

0.652847882

0.616564527

0.688701009

0.685077086

0.677712519

0.670540499

346

0.64749631

0.603703793

0.693652393

0.685901134

0.671054531

0.667105867

347

0.636562148

0.599764383

0.678201305

0.680230749

0.662847725

0.661717776

348

0.643236215

0.604302233

0.679852481

0.67251794

0.656248486

0.657676837

349

0.640973127

0.613965938

0.675675708

0.666682015

0.665942297

0.660034052

350

0.637281567

0.607143655

0.670385705

0.676808076

0.655081792

0.659908595

351

0.646600747

0.602071355

0.666624407

0.666859477

0.657862451

0.658502301

352

0.640837928

0.595797374

0.677917235

0.660348845

0.659032686

0.660174297

353

0.640178802

0.604447899

0.683071267

0.672725931

0.668688764

0.658344137

354

0.643672724

0.59717748

0.682664092

0.669672691

0.656946335

0.654177998

355

0.636845532

0.599731677

0.675412594

0.672892943

0.657042071

0.65165466

356

0.629589951

0.602254042

0.669941846

0.661581076

0.649528436

0.644018439

357

0.632261927

0.578343574

0.662890774

0.660941212

0.652529824

0.644018516

358

0.628107306

0.568984535

0.660804634

0.659455397

0.646102327

0.647908558

359

0.60651037

0.577252516

0.654078721

0.646666027

0.640831653

0.640205611

360

0.612716807

0.572272005

0.657080775

0.651767596

0.629280195

0.639590846

361

0.619847168

0.575405962

0.654678842

0.640859902

0.636606244

0.642093033

362

0.61727699

0.586610931

0.659639557

0.658177142

0.648690168

0.641453824

363

0.616285663

0.573558451

0.652192007

0.650740769

0.628245469

0.638855219

364

0.626221272

0.595019695

0.657808138

0.644416801

0.641024052

0.635364669

365

0.620685086

0.589522875

0.658299609

0.656351491

0.643386215

0.636563117

366

0.61809636

0.578012178

0.651705627

0.644449145

0.635540649

0.630985417

367

0.623596767

0.581979039

0.651940162

0.64706723

0.638606784

0.631927543

368

0.616151966

0.561953127

0.649382251

0.646392706

0.636196335

0.63153405

369

0.621138165

0.557220858

0.649018363

0.650910985

0.637850164

0.63102922

370

0.600483516

0.556210942

0.644761761

0.642982849

0.62582224

0.62003803

371

0.59276026

0.552911099

0.640016008

0.631178465

0.622235427

0.617220926

372

0.593147865

0.554177168

0.631856026

0.635382424

0.61693815

0.621058239

373

0.590340759

0.55007655

0.628890505

0.614968133

0.608103634

0.615268626

374

0.584943783

0.533906973

0.623694691

0.622661151

0.59973259

0.609904904

375

0.595784526

0.539406471

0.624678442

0.625137494

0.610139896

0.608203073

376

0.578553019

0.52745636

0.626651187

0.624811728

0.610781407

0.604959117

377

0.575881316

0.524289291

0.62240805

0.627673615

0.602300894

0.600577942

378

0.561744204

0.51912926

0.620182509

0.611609995

0.589484275

0.600014796

379

0.574488828

0.536456465

0.622007984

0.620770347

0.594513191

0.598571526

380

0.574543319

0.538486863

0.613668272

0.604776911

0.592016383

0.596279318

381

0.574904932

0.540168383

0.617469968

0.618813349

0.596922532

0.60243711

382

0.575776608

0.530777555

0.615161873

0.601939016

0.588449739

0.594689561

383

0.583464541

0.536175648

0.620147877

0.60345079

0.596724107

0.593586578

384

0.578864728

0.528911413

0.609187308

0.609718522

0.599228616

0.587035227

385

0.566138947

0.519124844

0.597614759

0.609002629

0.585980059

0.57771974

386

0.549496178

0.514909189

0.5902328

0.586160505

0.572021829

0.5748271

387

0.551348977

0.586479256

0.580961629

0.568486331

388

0.554351766

0.59337917

0.583518044

0.563606878

389

0.57898305

0.56991143

390

0.585796718

Data for Figure 4, Panel B: Calendar Time

Intraday Bin (min)

MPV3

MPV5

MedRV

TRV

TBV

RMinRV

1

4.370750452

4.740400589

4.415640969

3.535681778

3.979219486

4.711224092

2

5.096028845

5.495976812

4.804845927

4.745766509

4.832355928

4.975000362

3

4.978970841

5.420605325

4.619514077

4.631032096

4.678912041

4.732803131

4

4.799709973

5.17386223

4.379182316

4.356979799

4.500051249

4.438126301

5

4.530036308

4.858448561

4.14345153

4.131012902

4.276160913

4.176638391

6

4.283695514

4.591605905

3.878935417

3.973897984

4.051173738

3.90299568

7

4.024179101

4.396814438

3.628113564

3.735934186

3.850716237

3.692340596

8

3.750360485

4.169432076

3.391783842

3.487498181

3.570746266

3.507736574

9

3.547199992

3.852017863

3.275500902

3.309199003

3.408131381

3.301734789

10

3.435392502

3.71526857

3.127165843

3.160676964

3.262484033

3.151696797

11

3.221545708

3.654800483

2.994680095

3.070634345

3.179994132

3.077771437

12

3.119706194

3.534587789

2.806663306

2.982788255

2.955143091

3.00533546

13

3.052378701

3.342916458

2.774808232

2.727407881

2.832205027

2.897035575

14

3.077459544

3.294895303

2.838819906

2.753292768

2.922409015

2.831341638

15

2.949673678

3.1669668

2.733727096

2.785069294

2.883728196

2.747802727

16

2.797483483

3.007793704

2.590304132

2.700423165

2.726459812

2.620995914

17

2.675331151

2.939417307

2.445761737

2.538486359

2.585918156

2.540147455

18

2.602733557

2.861665958

2.414334814

2.454487657

2.509338452

2.472094033

19

2.542286366

2.811739604

2.338501018

2.334751739

2.443253819

2.391561173

20

2.503801503

2.701412923

2.305351827

2.365098246

2.415249911

2.354452072

21

2.426831585

2.614674893

2.237223888

2.296480598

2.362183616

2.276254735

22

2.371027858

2.502593783

2.211096369

2.290120889

2.296967587

2.22519356

23

2.255507474

2.457271065

2.091852719

2.127953495

2.221585983

2.16023039

24

2.206364606

2.445704144

2.068136238

2.159790685

2.162007109

2.163673539

25

2.161942307

2.391539393

2.003535911

2.0041421

2.070441371

2.124264104

26

2.204480355

2.391846672

2.055116556

2.026225463

2.087597799

2.112079367

27

2.191184872

2.57954148

2.047124717

2.045189815

2.136063933

2.267105526

28

2.181115812

2.68430498

2.041648265

2.113279677

2.129903036

2.34009

29

2.391360451

2.709440624

2.199925907

2.006455767

2.052041914

2.386400507

30

2.581795829

2.801785126

2.402063826

1.989506043

2.42149054

2.42929785

31

2.747171617

2.782025889

2.540069246

2.777379988

2.72691115

2.419613627

32

2.428764192

2.486295596

2.254657623

2.361368005

2.380063324

2.218938595

33

2.166819282

2.296224432

2.058688504

2.179409365

2.199729418

2.068056342

34

2.068993516

2.181331153

1.955120374

2.03770089

2.005805106

1.958994012

35

1.948671382

2.077446174

1.842091423

1.878158767

1.920121324

1.892198487

36

1.952106241

2.030273877

1.835218458

1.924256618

1.893887435

1.848410292

37

1.836201822

1.960480529

1.755988072

1.775506826

1.81981658

1.777629081

38

1.791771935

1.885543681

1.712846922

1.756516813

1.762901122

1.741165708

39

1.766896988

1.854657737

1.683575604

1.69553675

1.717414982

1.701152201

40

1.711521138

1.792912397

1.627309445

1.661569384

1.709529547

1.667210265

41

1.682804126

1.755493502

1.606474064

1.678349997

1.663808923

1.62545956

42

1.622134576

1.72885357

1.559098386

1.575509843

1.593132381

1.607758142

43

1.612800268

1.719393268

1.549012345

1.570859849

1.590926113

1.59352112

44

1.606544074

1.68969764

1.556654264

1.550283059

1.557438302

1.586746019

45

1.592764237

1.669273042

1.537064179

1.514268185

1.583852311

1.544086307

46

1.597242273

1.62137349

1.52920635

1.590205838

1.579106289

1.518194938

47

1.542682992

1.575589453

1.478651983

1.502538993

1.534463956

1.493303363

48

1.46550299

1.520567672

1.416526175

1.497189641

1.486586059

1.465287889

49

1.435098027

1.501706026

1.376549253

1.398648936

1.388144883

1.432167274

50

1.44641766

1.484799233

1.390037165

1.364944228

1.425906259

1.432996748

51

1.444817275

1.486042831

1.401327028

1.447113817

1.44933454

1.432743518

52

1.412934681

1.444165811

1.383049929

1.408000378

1.403665433

1.396922385

53

1.389498283

1.455212021

1.373975201

1.370779923

1.382961924

1.383621329

54

1.371236173

1.428972124

1.348144884

1.380255182

1.385369707

1.362941047

55

1.371249008

1.397172573

1.324514796

1.352974657

1.344603824

1.348613423

56

1.34215223

1.367408764

1.311052073

1.298136357

1.329062174

1.315879795

57

1.360959721

1.457409446

1.322952546

1.352956457

1.347315486

1.40042066

58

1.325460798

1.506811944

1.294514756

1.281602506

1.322513366

1.424970329

59

1.438528389

1.558250365

1.3717351

1.323137357

1.307257245

1.472509521

60

1.521076745

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0.91060029

386

0.882363428

1.094630465

0.855879891

0.887131294

0.853703314

1.004989555

387

0.917457014

0.888657438

0.794851879

0.842269623

388

1.122619896

1.080891943

0.854240014

0.935431661

389

0.96772791

1.166262093

390

1.253702191

To systematically assess the empirical relevance of intra-day
volatility fluctuations, and in particular the diurnal volatility
pattern, we split our equity sample into sub-groups consisting of
stock-day combinations representing the top and bottom deciles with
respect to a simple scale-free proxy for the intraday variation in
volatility. This proxy, denoted
(volatility-of-volatility), is constructed by splitting the trading
day into 26 blocks, and then obtaining the median 30 second
absolute returns within each block after first having eliminated
all zero returns. Our proxy is then defined as
the coefficient of variation (standard deviation divided by mean)
of these 26 medians.27 Importantly, the measure is designed to pick up any significant
variation in intra-day volatility and not just the commonly
occurring U-shape. In the following section, we provide signature
plots that broadly support our interpretation of the simulation
evidence.

6.3 Signature Plots for Integrated Power
Variation

Panel A of Figure 5 provides
signature plots for the IV estimates averaged across all days and
stocks. They are plotted as a function of the sampling frequency
implied by the size of the pre-averaging window and the intensity
of the transactions (in tick time). The RV estimator is included as
a reference point and, as expected, lies above the jump-robust IV
estimators which are bundled closely together across most of the
frequencies.28 The main outliers are MPV5 at
frequencies higher than 120 seconds and QRV at frequencies lower
than 90 seconds. At high frequencies, the drop in MPV5 and, to a
lesser extent, MPV3 may be explained by the presence of zero
returns, due to discreteness, which has a pronounced impact on
these estimators. Consistent with this explanation, RV is the only
estimator not to display a tendency to fall off at the highest
frequencies. At lower frequencies, the striking downward bias of
QRV is in line with the evidence from the simulation experiments.
While all the jump-robust measures feature downward sloping
signature plots, QRV is, by far, subject to the most significant
distortion.

The IQ estimators, shown in Panel A of Figure 6,
display a similar but more pronounced pattern of decline at the
lowest sampling frequencies, consistent with the simulation
scenarios that incorporate time-varying volatility. In particular,
the estimators are roughly ordered by block size, with the 5-block
estimators at the bottom and the 3-block estimators at the top. The
truncation based IQ estimators, although nominally based on a short
block, are disproportionately impacted by time variation in
volatility due to the wide window necessary for determining the
truncation thresholds, explaining the relatively sharp decline in
their signature plots. We do not include an IQ analogue to the QRV
estimator as it displays downward biases that are an order of
magnitude larger than for the others.

The signature plot for the
ratio in Panel A of Figure 7 displays a
relatively flat but distinct monotonically declining pattern. This
suggests that the microstructure effects, afflicting estimators at
the highest frequencies, cancel out quite effectively at moderate
frequencies consistent with the simulation evidence. The
multi-power variation estimators display clear abnormalities at the
highest frequencies, primarily due to an imperfect offset of the
zero returns in the denominator. At sampling frequencies of 60
seconds or lower, however, all the estimators are tightly clustered
and downward sloping in accordance with the findings from the
simulation scenarios with time-varying volatility.

To further explore the impact of time-varying volatility, Panels
B and C of each figure depict signature plots, respectively, for
the top and bottom deciles with respect to our
volatility-of-volatility proxy, , across the
combined stock-day sample. It is evident that both the IQ and IV
signature plots on high VoV days are dramatically more steeply
sloped than on low VoV days, corroborating the hypothesis that the
slope is caused primarily by time-varying volatility due to factors
like the intraday U-shape pattern, volatility jumps, and volatility
bursts associated with news effects. For the ratio statistic in
Figure 7, the signature
plot for low VoV days is essentially flat, with the exception of
the multi-power estimators, when viewed on the scale of the high
VoV days, highlighting the fact that time variation in volatility
also has a significant impact on the ratio statistic.

While the above evidence makes a fairly compelling case that VoV
has a pronounced impact on the slope of the signature plots, it is
based on grand averages across stocks and days and does not allow
for formal statements about statistical significance. To provide a
more rigorous analysis, controlling for the potential impact of a
few outlier stocks, we have also run a series of panel regressions
with stock fixed effects, capturing the average effect of VoV on
the slope of the signature plot for each individual stock. These
regressions confirm the strong significance of VoV on the slope of
the signature plots.29

Given the accumulated evidence we consider a sampling frequency
around 90-120 seconds as a sensible choice for inference about IV
and jump tests based on the
ratio across most of the
estimators. At higher frequencies, the microstructure effects start
impacting the estimators quite severely, while employing lower
frequencies entails a significant loss of efficiency and ultimately
also generates a severe downward bias, especially for days with
turbulent market conditions when volatility-of-volatility
fluctuates greatly. Furthermore, among the set of estimators we
consider, the RNT estimators, and , appear to possess advantages in
both efficiency and robustness.

Figure 5: Average Estimates of IV Across the DJ30 Stocks Between January 1, 2005 and May 31, 2007

We provide signature plots for the mean of each pre-averaged estimator of IV as a function of pre-averaging window size matching the sampling frequency (measured in seconds on the x-axis). Panel A plots the mean across all days.
Panel B plots the mean across the top 10% days with respect to intraday variation in volatility. Panel C plots the
mean across the bottom 10% days with respect to intraday variation in volatility. Intraday variation in volatility is
measured by the V oV measure of volatility of volatility described in Section 6.

Data for Figure 5, Panel A: Mean (x 10^4) of IV For All Days

Frequency (sec)

RV

MPV3

MPV5

MedRV

TRV

TBV

QRV

RMinRV

30

1.251884583

0.958551049

0.794837137

1.050386554

1.108841871

1.021009036

1.050368286

1.011301061

45

1.240427788

1.015900953

0.905233404

1.061270742

1.116815868

1.051249276

1.048687263

1.0373291

60

1.245180899

1.054981825

0.970684965

1.081438625

1.133234936

1.078672784

1.057689488

1.061914918

75

1.252532164

1.082041457

1.012162791

1.100383368

1.147583156

1.099965949

1.065566161

1.081304584

90

1.25903186

1.101218647

1.039562458

1.114048418

1.158290921

1.114885241

1.070461982

1.095103259

120

1.2680465

1.123873114

1.070614799

1.132082931

1.171693346

1.132898599

1.072164372

1.111266901

150

1.272537466

1.134766757

1.084669682

1.139628165

1.177157883

1.140914767

1.067865734

1.117304693

180

1.274062492

1.139652713

1.091031384

1.142783284

1.178085612

1.142917754

1.061406503

1.118967025

225

1.272713985

1.139434501

1.092115561

1.14171477

1.174619809

1.140426198

1.048426017

1.11616094

300

1.264602949

1.130578613

1.083323964

1.131335189

1.160997601

1.126193892

1.026726498

1.103180975

450

1.238713518

1.103201283

1.053442306

1.102535903

1.127560864

1.087865084

0.989489818

1.071352117

600

1.212630859

1.07727044

1.024563562

1.075803988

1.098021964

1.052253891

0.968514422

1.0424071

Data for Figure 5, Panel B: Mean (x 10^4) of IV for the Top 10% Days in Terms of Intraday Variation of Volatility

Frequency (sec)

RV

MPV3

MPV5

MedRV

TRV

TBV

QRV

RMinRV

30

1.791308526

1.28832395

1.109858253

1.379652254

1.474582587

1.357855589

1.385674883

1.332443208

45

1.808395937

1.36966899

1.233389255

1.425229399

1.509050078

1.417710152

1.393857513

1.382516833

60

1.825636059

1.425067337

1.312078123

1.4585819

1.531895237

1.459318265

1.399377542

1.416900981

75

1.839286468

1.455663491

1.356757493

1.479419676

1.544269167

1.485873165

1.399885101

1.435761221

90

1.849269302

1.476240167

1.382050553

1.489296872

1.551836014

1.498829328

1.394505684

1.445393125

120

1.864338254

1.496091624

1.409454643

1.500399676

1.557248506

1.513628464

1.379177435

1.451926121

150

1.873893928

1.501409369

1.412835636

1.494248369

1.551952373

1.515325927

1.359331111

1.442332755

180

1.879515242

1.500622253

1.411383763

1.489619882

1.541556205

1.507034983

1.34193356

1.433314958

225

1.88292431

1.482199717

1.395997252

1.472438277

1.523515411

1.488578865

1.307610203

1.413496149

300

1.878663654

1.451867085

1.367440507

1.439637528

1.48313302

1.450070889

1.262404469

1.380603429

450

1.849924495

1.40630547

1.315660776

1.388536935

1.411109831

1.376042037

1.188980342

1.332148809

600

1.816754692

1.374067481

1.277159656

1.352763392

1.358194549

1.312988669

1.158099485

1.292541786

Data for Figure 5, Panel C: Mean (x 10^4) of IV for the Bottom 10% Days in Terms of Intraday Variation of Volatility

Frequency (sec)

RV

MPV3

MPV5

MedRV

TRV

TBV

QRV

RMinRV

30

1.064453186

0.808401941

0.63944721

0.923949842

0.968292186

0.880766271

0.930248805

0.876132417

45

1.032398946

0.85268469

0.744624692

0.906240363

0.952690049

0.891556499

0.909594605

0.883767604

60

1.026416935

0.882320488

0.804331714

0.91103945

0.957266407

0.906735756

0.911336901

0.897698073

75

1.028838898

0.905278379

0.843978628

0.923324475

0.967302336

0.922738742

0.920267602

0.912815121

90

1.0317358

0.921134382

0.870431614

0.933955688

0.974988916

0.933904004

0.924939003

0.924588799

120

1.036663109

0.941568242

0.902112339

0.948157048

0.986868364

0.948836198

0.931165084

0.940793712

150

1.038761362

0.95147137

0.917081798

0.956034978

0.992809489

0.956542772

0.931747254

0.947183656

180

1.038800034

0.957645367

0.925342794

0.96098025

0.995418964

0.960751063

0.92995458

0.951781305

225

1.037083671

0.961733579

0.932318162

0.96456372

0.995250643

0.961887961

0.923971107

0.954821329

300

1.031355213

0.959019955

0.931353631

0.961688764

0.989846395

0.955404304

0.912235765

0.950804045

450

1.013458778

0.947926465

0.918609213

0.947701279

0.972184923

0.934043604

0.885822933

0.933863524

600

0.99400008

0.929011307

0.900639918

0.928322869

0.951593867

0.909432162

0.870820867

0.913544516

Figure 6: Average Estimates of √IQ Across the DJ30 Stocks Between January 1, 2005 and May 31,
2007

We provide signature plots for the mean of each pre-averaged estimator of √IQ as a function of pre-averaging
window size matching the sampling frequency (measured in seconds on the x-axis). Panel A plots the mean across
all days. Panel B plots the mean across the top 10% days with respect to intraday variation in volatility. Panel C
plots the mean across the bottom 10% days with respect to intraday variation in volatility. Intraday variation in
volatility is measured by the V oV measure of volatility of volatility described in Section 6.

Data for Figure 6, Panel A: Mean (x 10^4) of sqrt(IQ) For All Days

Frequency (sec)

sqrt(MPQ3)

sqrt(MPQ5)

sqrt(MedRQ)

sqrt(TRQ)

sqrt(TBQ)

sqrt(RMinRQ)

30

1.442376695

1.271094357

1.50213521

1.504948667

1.444731101

1.407577956

45

1.444610388

1.307127859

1.485826066

1.49324215

1.448247294

1.408467014

60

1.452406498

1.331527593

1.480005654

1.489189791

1.454337115

1.412584774

75

1.456050694

1.345995836

1.480198261

1.483994876

1.459416869

1.414846807

90

1.458908677

1.353859658

1.474429668

1.477593141

1.459394652

1.413456046

120

1.451633657

1.35302455

1.462847805

1.460877787

1.450830461

1.402183023

150

1.437650566

1.341985334

1.443333611

1.440123597

1.435805524

1.383074686

180

1.420556028

1.326830613

1.423955646

1.419035361

1.417242032

1.363768566

225

1.390721105

1.299967575

1.395086586

1.387299716

1.388898641

1.33589928

300

1.345483541

1.256945533

1.347919167

1.335407628

1.337876703

1.288199082

450

1.269947515

1.179696407

1.271543387

1.250640528

1.249169191

1.211242228

600

1.214550642

1.121440971

1.215479447

1.188109624

1.18259545

1.153606787

Data for Figure 6, Panel B: Mean (x 10^4) of sqrt(IQ) For the Top 10% Days in Terms of Intraday Variation of Volatility

Frequency (sec)

sqrt(MPQ3)

sqrt(MPQ5)

sqrt(MedRQ)

sqrt(TRQ)

sqrt(TBQ)

sqrt(RMinRQ)

30

2.414740215

2.131835896

2.529935902

2.546686961

2.468925337

2.366451326

45

2.420568695

2.176680926

2.50887977

2.511243842

2.473547662

2.362449527

60

2.429457688

2.209532142

2.475168122

2.456280306

2.456847426

2.339163034

75

2.399010398

2.200776601

2.443093202

2.39928646

2.43300215

2.303167315

90

2.378259469

2.181394988

2.389039727

2.350623616

2.389946912

2.262629619

120

2.312344377

2.125137108

2.312115428

2.264330651

2.320651894

2.180939663

150

2.247092381

2.057203767

2.221097096

2.180940004

2.252617282

2.091878004

180

2.186278118

1.997357848

2.156025381

2.107443178

2.179336298

2.022320003

225

2.074451936

1.896801638

2.057565038

2.010745134

2.080771114

1.92552261

300

1.949026038

1.782661999

1.932553109

1.867569482

1.939285464

1.806456873

450

1.801909048

1.629143347

1.779328859

1.668547958

1.734506235

1.660187785

600

1.717754237

1.535824664

1.691467327

1.54234234

1.592346649

1.570102974

Data for Figure 6, Panel C: Mean (x 10^4) of sqrt(IQ) For the Bottom 10% Days in Terms of Intraday Variation of Volatility

Frequency (sec)

sqrt(MPQ3)

sqrt(MPQ5)

sqrt(MedRQ)

sqrt(TRQ)

sqrt(TBQ)

sqrt(RMinRQ)

30

1.090740691

0.946622293

1.139040823

1.141013853

1.089748749

1.061179101

45

1.07110994

0.965648165

1.103721548

1.119139565

1.072358894

1.047801578

60

1.071849801

0.982760021

1.096716227

1.118325466

1.074127235

1.049474111

75

1.077733714

0.998254846

1.09861362

1.121553326

1.082446844

1.056759469

90

1.08192871

1.009411932

1.100501898

1.122008225

1.085647361

1.060421738

120

1.085857229

1.021904627

1.096876864

1.122263116

1.087921338

1.064571332

150

1.083122395

1.024420811

1.091681302

1.117110224

1.086362856

1.060617879

180

1.080635847

1.024169381

1.087006936

1.111299634

1.084712838

1.058277823

225

1.072625041

1.020861563

1.079842314

1.098481188

1.074911694

1.054189024

300

1.053372014

1.00387738

1.058810539

1.075018636

1.052173614

1.033618461

450

1.022146184

0.972097321

1.022660943

1.033864489

1.009762647

0.997645627

600

0.984909963

0.937265354

0.987750397

0.995117265

0.970799129

0.959921452

Figure 7: Average Estimates of √IQ/IV Across the DJ30 Stocks Between January 1, 2005 and May
31, 2007

We provide signature plots for the mean of each pre-averaged estimator of √IQ/IV as a function of
pre-averaging window size matching the sampling frequency (measured in seconds on the x-axis). Panel A plots the
mean across all days. Panel B plots the mean across the top 10% days with respect to intraday variation in volatility.
Panel C plots the mean across the bottom 10% days with respect to intraday variation in volatility. Intraday variation
in volatility is measured by the V oV measure of volatility of volatility described in Section 6.

Data for Table 7, Panel A: Mean of sqrt(IQ)/IV For All Days

Frequency (sec)

sqrt(MPQ3)/MPV3

sqrt(MPQ5)/MPV5

sqrt(MedRQ)/MedRV

sqrt(TRQ)/TRV

sqrt(TBQ)/TBV

sqrt(RMinRQ)/RMinRV

30

1.507107116

1.688520391

1.404610311

1.333432815

1.39682524

1.372414188

45

1.415373352

1.48084126

1.377412284

1.315776639

1.358444536

1.339494557

60

1.365901129

1.384631033

1.350467704

1.29561673

1.330886944

1.313878287

75

1.333151781

1.332468752

1.328276659

1.276388978

1.310060192

1.292631281

90

1.309737645

1.29804305

1.30761283

1.25971896

1.292462826

1.275175229

120

1.274756732

1.25391734

1.275058403

1.231385714

1.263612388

1.245655383

150

1.249296936

1.224467131

1.249958514

1.208625869

1.241402007

1.222468807

180

1.228192509

1.201834695

1.22944161

1.190278298

1.223588791

1.203707929

225

1.203203523

1.176295435

1.205695507

1.167386557

1.201719466

1.181676008

300

1.173216794

1.145827553

1.176052292

1.138743558

1.173502872

1.154103509

450

1.134413428

1.107867389

1.1386597

1.101003192

1.136386898

1.118946506

600

1.11122577

1.083654651

1.115551977

1.076910652

1.114104639

1.096331805

Data for Table 7, Panel B: Mean of sqrt(IQ)/IV for the Top 10% Days in Terms of Intraday Variation of Volatility

Frequency (sec)

sqrt(MPQ3)/MPV3

sqrt(MPQ5)/MPV5

sqrt(MedRQ)/MedRV

sqrt(TRQ)/TRV

sqrt(TBQ)/TBV

sqrt(RMinRQ)/RMinRV

30

1.869772906

1.996228824

1.796450552

1.696402124

1.792326917

1.749964552

45

1.758004101

1.805429889

1.723958424

1.632565288

1.71478982

1.682568868

60

1.687659347

1.697808089

1.665577579

1.572967575

1.652188984

1.625985966

75

1.627424848

1.620863016

1.619109269

1.523349664

1.603359539

1.576199203

90

1.582478584

1.564781002

1.573784151

1.485077037

1.560275903

1.536602538

120

1.509960938

1.480947096

1.504130427

1.421401529

1.493874943

1.466579861

150

1.455961758

1.421286795

1.450322205

1.372759061

1.446281291

1.411986319

180

1.41476323

1.373692593

1.411000922

1.333901413

1.406450929

1.371972336

225

1.361373786

1.321218166

1.359959206

1.288107781

1.35865779

1.322928517

300

1.302423076

1.261113023

1.299854272

1.232724758

1.301405497

1.268472521

450

1.233709268

1.194729351

1.236934173

1.162867292

1.225713283

1.202619253

600

1.199779429

1.157911717

1.19962003

1.120995303

1.182343855

1.170356688

Data for Table 7, Panel C: Mean of sqrt(IQ)/IV for the Bottom 10% Days in Terms of Intraday Variation of Volatility

Frequency (sec)

sqrt(MPQ3)/MPV3

sqrt(MPQ5)/MPV5

sqrt(MedRQ)/MedRV

sqrt(TRQ)/TRV

sqrt(TBQ)/TBV

sqrt(RMinRQ)/RMinRV

30

1.390598436

1.66766297

1.23369788

1.180117752

1.248479554

1.218386657

45

1.289474933

1.400654313

1.228630236

1.181470229

1.216147175

1.198347103

60

1.240935776

1.284651078

1.217473453

1.177749366

1.200256028

1.183473185

75

1.214108691

1.229014611

1.205639107

1.171032638

1.190426406

1.173717965

90

1.197194467

1.197478017

1.195587597

1.162899062

1.18087963

1.164091418

120

1.172654296

1.161143029

1.174362496

1.148153688

1.163542539

1.147244041

150

1.155875825

1.139176304

1.15951379

1.134994194

1.150328581

1.135334708

180

1.143499694

1.124863049

1.145710064

1.125046599

1.141698091

1.125265033

225

1.126905012

1.107731499

1.132027953

1.110534543

1.12896346

1.114049269

300

1.108840734

1.088584894

1.113738203

1.091127156

1.111981387

1.095780815

450

1.083647284

1.065574076

1.085655326

1.066502664

1.088201084

1.074401761

600

1.066042116

1.047476475

1.069893784

1.049288809

1.074252286

1.057151672

7 Conclusion

We provide a first in-depth look at robust estimation of
integrated quarticity (IQ) based on high frequency data. The
sensitivity of many existing IQ estimators to pervasive data
irregularities inspires us to introduce a novel set of jump-robust
estimators that are defined in terms of order statistics of
suitable return functionals and generalize the existing nearest
neighbor truncation estimators of ADS (2012). This new class of
robust neighborhood truncation (RNT) estimators can be designed to
enhance the robustness properties vis-à-vis microstructure
noise features of the data as well as reducing the finite sample
sensitivity to outliers. The identical principle can more generally
be applied also to other consistent estimators to enhance jump- and
noise-robustness. We find that the novel RNT estimators outperform
existing estimators by a considerable margin in terms of finite
sample efficiency in estimating the key ratio of . This quantity is extremely useful for robust
inference regarding IV and for testing for price jumps.

In the empirical implementation, we emphasize the importance of
appropriate filtering for gross violations of the particular null
hypothesis associated with a given estimation procedure. In
particular, we apply a novel functional filtering scheme for local
power variation estimators, which generalizes truncation of
individual returns to truncation of return functionals and is easy
to apply for a broad range of popular estimators. By invoking this
approach at the level of the local power variation estimators, the
threshold can be set very conservatively, thereby avoiding
systematic biases arising from aggressive truncation and thus
eliminating the need for ex-post bias correction. We also emphasize
the use of pre-averaging based on a wide pre-averaging window. This
allows for important robustness to extreme outliers, like the
so-called bounce-backs, and it simplifies the associated
distribution theory as the impact of noise vanishes
asymptotically.

The unifying theme behind the new class of estimators, as well
as the universal filtering procedure applied to them, is to operate
directly on the functional space of local power variation estimates
instead of restricting attention to the underlying individual
returns. In fact, we may view the latter as a special case, arising
from a block size of unity. Combining the functional filtering with
the novel RNT class of estimators enables efficient inference in an
extensive simulation design and generates supportive evidence from
an empirical application using the Dow Jones 30 stocks. Overall,
the study provides a set of new guidelines for the construction of
practical robust and efficient estimation and inference regarding
IV and IQ.

Hautsch, N. and M. Podolskij (2010). Pre-averaging based estimation of quadratic variation in
the presence of noise and jumps: Theory, implementation, and empirical evidence. Journal of
Business and Economic Statistics forthcoming.

A Proofs of Propositions

This appendix
provides proofs of Proposition 1 and related results for the
general case of estimating the integrated power variation of order
, where is a positive and
even integer. The proofs are initially given for the MinPV and
MedPV type estimators and subsequently shown to extend to the RNT
estimators in Lemma 9 below.

The
(integrated) power variation, , is formally
defined as,

(3)

Obviously,
corresponds to the theoretical quantity
relevant for the and
estimators while refers to the integrated
variance underlying the and estimators. The higher order integrated power variation
estimators are less commonly used but do appear in the recent
literature. For example, is required to assess the
(asymptotic) precision of integrated quarticity
estimators.

A.1 Basic Setting

Let be the log price process following a Brownian
semimartingale

(4)

where is a locally bounded and predictable process and
is adapted, cadlag and bounded away
from zero. Without loss of generality, we further assume that the
functions
are uniformly bounded and
a.s.30 The extension allowing for finite
activity jumps in is dealt with Section A.6 below.

When discussing
central limit theorems (CLTs) we require in addition that the
volatility process follows a generalized Itô
process:

where is locally bounded and predictable and
are cadlag and
the Brownian motions are uncorrelated. We
impose, without loss of generality, that the functions
and
are uniformly bounded as well as
and
a.s. We
further note that, when the volatility process
satisfies Assumption
A1, then the power variation process,
, also conforms to
this general characterization.

We assume
is observed at evenly
spaced time points spanning the interval .
Below, we denote these observations by ,
, and the associated log-returns
by
,
. The proofs involve sequences
of standardized return observations and corresponding approximating
sequences for which volatility is fixed across one or more returns.
Hence, we introduce non-overlapping blocks of returns for which the volatility process is
constant. We assume we have such blocks in
the sample. Consequently, we define the
quantities,

(5)

(6)

where
indicates the integer
part of an expression. Hence, for each of the
return blocks, corresponding to
, the volatility remains fixed
at the value it attains at the beginning of the
block.

A.2 The Min and Med Power Variation
Estimators

Let be a fixed positive even integer and let
be
given by,

where the scaling constant takes the form,

(7)

and

For example, we
have
and
.

Similarly, we
define the median-based function
and scaling factors,

where

and

In this case,
and
.

For any even
positive integer, , we define the nearest
neighbor truncation estimators of the 'th order
power variation by,

MinPV

MedPV

For the cases
of primary interest, i.e., and , these estimators are identical to the and estimators introduced in Section
2.1. Specifically, we have

MinPV MinRV
MedPV
MedRV

MinPV MinRQ
MedPV
MedRQ

A.3 Additional Notation and Preliminary
Results

We provide a
detailed proof of the results in Propositions 1 and 2 concerning
the
MinPV estimator. The proofs for
MedPV may be derived similarly. Moreover, we
henceforth consider a fixed even, positive integer, ,
so the
function is uniquely
defined. We refer to it simply as
below.

First, we
observe that, for any bivariate vectors,
and
, we have the following
useful bound,

(8)

and furthermore
that, except on the null set
, we
have

(9)

The proofs of
Propositions 1 and 2 revolve around the
sequences,

and

Since
MinPV the
sequence is asymptotically equivalent to
our estimator, while is an approximating sequence as, for large
.

For any
adapted, integrable, -dimensional cadlag process,
, and for
, we define the
expectation conditional on information at time
:

(10)

A useful
implication of our ability to focus on the case with uniformly
bounded drift and volatility functions is that, using the
Burkholder-Davis-Grundy inequalities, we
have,

and

(11)

where
and
denotes a generic positive constant which we (with slight abuse of
notation) allow to take on disparate values in different
places.

We decompose
our estimators for the power variation, , into a
sum of conditional expectations and the associated martingale
difference sequences:
and
where,

When
we will use the shorthand
and similarly for the individual pieces
and . These
definitions allow us to decompose the main
estimator:

(12)

Consistency of
can then be obtained by showing
consistency of the estimator applied to the approximating Brownian
path with piecewise constant volatility (
) and then showing
that the difference
(the last two terms in (12)
above) is asymptotically negligible. This is what we do in Section A.4 below. To prove a CLT, we exploit a different decomposition
(similar to Mykland and Zhang (2007)), in which we show the CLT for
our estimator applied to an approximating Brownian motion for which
volatility is piecewise constant over blocks of length . We then proceed to show that the difference between the
original estimator and the estimator applied to the approximating
process is negligible. This analysis is carried out in Section A.5 based on the
decomposition:

(13)

A.4 Proposition 2: Consistency

We proceed by
analyzing equation (12) term by term
through a series of lemmas. For brevity, we focus on the features
that are specific to our estimator, while referring to proofs in
the extant literature when feasible. This also serves to highlight
the underlying structural similarities between our measure and previously proposed power variation
estimators and, in particular, and estimators.

Lemma 2Under the maintained assumptions we have,

(14)

Moreover, if Assumption (A1) holds we obtain,

(15)

Proof. First, note that

so we may write

(16)

The first sum in (16) tends to zero
in probability. To see this, note that the bound (8) implies the
following limit in -norm:

(17)

where the convergence (17), and thus also
convergence in probability, follows from the fact that
has finite quadratic
variation (since
is a cadlag semimartingale). In
addition, since
is uniformly
bounded and cadlag, the pointwise dominated convergence of
for
follows and Lebesgue's
theorem yields

(18)

Together (17) and (18) imply
which establishes (14). To show
(15)
we need the stronger assumption (A1). Define the sequence of
independent standard normals
, then
Assumption (A1) yields

(19)

sincece
is an even function of the Brownian path
. Now the
property (9) yields

(20)

This ensures that the first term in (16) is
asymptotically negligible, even when scaled up by . Hence, the remaining task is to show,

However, this is analogous to the common task of showing that

in the IV literature and the method of proof is, by now, well
established; see, e.g., BNGJPS where the result is shown in a
general setting (allowing for infinite activity jumps) of which the
current framework is a special case. A more intuitive and detailed
exposition is provided by Barndorff-Nielsen, Graversen, Jacod, and
Shephard (2006), henceforth BNGJS.

where we have defined the function
. This formulation maps directly
into the setting of BNGJPS where the results of this lemma are
proven in a more general setting and for a generic function subject to regularity conditions. In
particular, our function trivially satisfies the
continuous differentiability and polynomial growth conditions
necessary for the applicability of their analysis. An accessible,
albeit lengthy, account of the steps of the argument may be found
in BNGJS (2006, pp. 713-719). So while this proof is quite
involved, the above reformulation of the relevant inequalities, as
they arise within our specific setting, allows us to simply refer
to previously published work for the result.

Lemma 5Under the maintained assumptions, we have,

(25)

Moreover, we may strengthen this result further to
obtain,

(26)

Proof. In order to demonstrate the second result of
the lemma, which obviously implies the first, we define,

and we must then prove that,

This is a martingale difference sequence with respect to the
filtration
, so it
suffices to show,

as

Mimicking the type of steps undertaken in the proof of the previous
lemma, including application of the uniform bound on moments of
and
, we obtain,

As for the previous lemma, our reformulation of the task maps the
problem into the corresponding task in BNGJPS (2006) who prove a
corresponding lemma in a more general setting. A detailed account
of the requisite steps to complete this part of the proof may again
be gleaned from BNGJS (2006, pp. 704-706).

Taken together,
Lemma 2-3 and the
first parts of Lemma 4-5 simply the
consistency of our estimator under the minimal maintained
assumptions. The second parts of Lemmas 4-5 are critical for
the proof of the central limit theorem below.

A.5 Proposition 3: The CLT

Lemma
6Under assumption (A1), we have

(27)

where the constant
for
.

Proof. Consider splitting the scaled
return observations into blocks, the
of which is the vector
. The corresponding vector of observations from the approximating
Brownian motion where volatility is held constant over the block is
. Next, define by
the block estimator of volatility:

(28)

We wish to apply Theorem IX.7.28 in Jacod and Shiryaev (2003) to
. Defining the
martingale difference sequence
we can write

(29)

The last equality follows from the fact that each term in the
second sum is centered and has bounded variance (given the uniform
bound on ). Thus the sum divided by
will tend to zero provided
.

We must now verify conditions (7.27)-(7.31) of Theorem IX.7.28.
First note that
so that condition (7.27) is trivially satisfied. Condition (7.28)
follows from the fact that

(30)

where the convergence in probability (and in fact a.s.) is a
consequence of the volatility process being cadlag and uniformly
bounded. Next, we turn to condition (7.29). Let
, then
, which follows from the fact that the variables
are centered and that
is an even function. Condition (7.30),
stating that
, follows straightforwardly from the fact that is uniformly bounded.

Finally, let
be a bounded
martingale orthogonal to (i.e. the covariation
). We want
to show that, for each block ,
. For
consider the martingale
difference sequence
. By the martingale representation theorem,
for some predictable process
. Therefore the processes
and
are
orthogonal and the product,
is again
a martingale which must then have mean zero. This verifies
condition (7.31) and Theorem IX.7.28 in Jacod and Shiryaev (2003)
states that as (and hence and
) tend to infinity:

(31)

Lemma 7Under the maintained assumptions, we have

(32)

Proof. Defining
, we note that
is a martingale difference sequence with respect to the filtration
. To show that
in probability, it therefore suffices (by Doobs inequality, e.g.
Revuz and Yor (1999)) to show that

(33)

By the bound of we have

(34)

where the last inequality follows from the uniform boundedness of
and Lebesgues theorem.

Importantly,
the specification of the volatility process in Assumption (A1) may
be extended to include finite as well as infinite activity jump
processes subject only to the regularity conditions stipulated in
BNGJPS. This follows from the fact that the only terms in (13) affected by the
inclusion of jumps are the terms
and
which map into the
corresponding terms in BNGJPS as outlined in the proofs above. As
such, the distributional results of the paper cover a wide range of
underlying return generating processes.

A.6 The Asymptotic Distribution under Jump Alternatives

Suppose now the
log price process is given as , where
is a Brownian semimartingale of the form
(4) while
is a finite activity jump process. We show
below that the above results continue to hold.31The key is that
, which follows readily from Levy's modulus of continuity theorem
for Brownian motion. This immediately yields:

Proposition 8When is a finite activity jump process,
the asymptotic distribution of the and
estimators applied to the processes
and are
identical.

Proof. As before, we deal only with the case as the case is analogous. On a
given realization of the path there is a finite number of jumps, so
(asymptotically) at most one of the terms
or
includes a jump. Therefore, each term in the estimator (up to a
normalizing constant) is

regardless of whether a (single) jump occurred or not over
.
Since only finitely many terms differ,

so neither consistency nor convergence in distribution is
affected by the presence of finite activity jumps.

A.7 Robust Neighborhood Truncation
Estimators

We consider the
family of robust neighborhood truncation (RNT) estimators on a
block of i.i.d.
returns,
. The estimator is
then constructed by taking the
quantile of
unbiased estimators of
on the block. Denoting these
primitive estimators by
, we
can write the RNT estimator as

where the
is a scaling
factor.

Lemma 9Let
be a positive even integer and assume that
the estimators
satisfy the conditions of Proposition 1 and Proposition 2, then the
robust neighborhood truncation estimator,
, defined in
(35) is consistent
for
and satisfies a CLT.

Proof. We need to verify the three
properties,(8)-(9) and
symmetry, of the function used in the
theorem are satisfied when
.
We deal with each condition in turn.

Clearly, if each primitive estimator
is symmetric, so is
. Moreover,
if each
satisfies a bound of the
type (8), so
will
as it is
simply an order statistic of such bounded functionals. Finally,
assume that each
satisfies (9). Except on
a null set, there exists a neighborhood around each m-tuple
, on which
for some
. Therefore it follows that also
satisfies
(9).

Remark 10Since the NT estimators
(up to a scale factor) essentially are a special case of RNT, the
lemma applies to these as well.

B Noise Robustness Properties of the
Ratio

The ratio
plays an important role for both
IV inference and jump attribution in finite samples. This section
extends the analysis of Huang and Tauchen (2005) to show that the
ratio has certain desirable robustness
features in the presence of microstructure noise. Following
Ait-Sahalia and Mykland (2005), we assume that the true price
process () is observed at discrete points in time with an independent stationary
(possibly autocorrelated) measurement error () that results in an MA error structure in observed
returns:

where

For simplicity, we
focus the discussion here on the (RV,RQ) pairing and
denote:

The presence of
microstructure noise produces a bias in both the IQ and IV
estimators of the form:

(35)

In the special
case where
is
serially uncorrelated and denoting the noise to signal ratio
, the
expressions above simplify to,

(36)

Under the null of
no jumps,
are asymptotically
unbiased and consistent and,

as

The downward
bias of the limiting ratio depends on the noise-to-signal ratio
and preaveraging of returns
therefore plays an important role in reducing and the associated distortions.32 Moreover, for
sufficiently pre-averaged returns, there is very little evidence of
serial correlation, as pointed out by COP (2010), and the serially
uncorrelated noise case considered above is therefore the
empirically most relevant case.33In finite samples,
this downward bias is further compounded by a pure Jensen
(concavity) effect as readily seen from the Monte Carlo results for
the Brownian motion scenario Path by path, the Cauchy-Schwartz
inequality of course implies that
must hold regardless of the noise structure or other
imperfections.

In the presence
of other deviations from the Brownian null, forming the ratio
may have a
stabilizing effect provided that the resulting distortion is
uniform and roughly proportional to squared returns since this will
lead to a cancelation in numerator and denominator. We see this
effect at work in the simulations with price discreteness but in
other instances, e.g., sparsity, it clearly fails. In cases
involving additive distortions such as jumps, there will be no
cancelation of biases and the ratio will tend to (in the case of
upward biases) diverge at high frequencies due to the scaling by N
in the numerator.

C and estimators based
on pre-averaged returns

An equivalent
definition with analytically more tractable expression is given
by:

(38)

where
is the
pre-averaging kernel. We further define
as the finite sample analog of the variance scaling factor
induced by
the pre-averaging kernel.34

Consider the
following sub-samples of non-overlapping
pre-averaged returns:

Let
and
,
denote the raw
and estimates
obtained on each sub-sample of pre-averaged returns. Then the
pre-averaged (and sub-sampled) estimators
and
for the full
set of pre-averaged returns
can
be defined as follows:

After
incorporating finite sample bias correction, these take the
following final form that we use in our pre-averaged implementation
of all estimators:

Consistency and
asymptotic normality are clearly preserved by pre-averaging and
sub-sampling, while noise-robustness improves. In particular,
bounce-backs are near perfectly annihilated given that adjacent
returns are subject to almost identical kernel weights. For further
details on pre-averaging please refer to Podolskij and Vetter
(2009) and Jacod, Li, Mykland, Podolskij and Vetter (2009) among
others.

Footnotes

** Torben G. Andersen, Kellogg School of Management, Northwestern University, 2001 Sheridan Road, Evanston, IL 60208, USA; NBER; CREATES; t-andersen@northwestern.edu.
Dobrislav Dobrev: Federal Reserve Board of Governors, 20th Street and Constitution Avenue NW, Washington, DC 20551, USA; Dobrislav.P.Dobrev@frb.gov.
Ernst Schaumburg: Federal Reserve Bank of New York, 33 Liberty Street, New York, NY 10045, USA; Ernst.Schaumburg@gmail.com.
We are grateful to two anonymous referees as well as the editor, Jun Yu, for comments. We also thank participants at the 2010 SETA Conference at Singapore Management University and the "Nonlinear and Financial Econometrics Conference: A Tribute to A. Ronald Gallant," Toulouse, France, May 2011, the NBER-NSF Time Series Conference, Michigan State University, September 16-17, 2011, the 5th International Conference on Computational and Financial Econometrics, London, December 17-19, 2011, the 5th Annual SoFiE Conference - Oxford-Man Institute, June 20-22, 2012, the North American Summer Meeting of the Econometric Society, Evanston, June 28 - July 1, 2012, along with Federico Bandi, Peter R. Hansen, Andrew Patton, Peter C. B. Phillips and Kevin Sheppard for comments on an earlier draft. Excellent research assistance was provided by Patrick Mason.
Andersen gratefully acknowledges financial support from the NSF through a grant to the NBER and by CREATES funded by the Danish National Research Foundation.
The views in this paper are solely those of the authors and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System, the Federal Reserve Bank of New York or of any other person associated with the Federal Reserve System. Return to text

2. Additional work regarding inference on
the jump and continuous components of the return variation is
developed in BNS (2004b, 2006), while general methodological
insights were provided by Jacod and Protter (1998). Subsequently,
market microstructure complications were addressed as part of the
drive to exploit tick level data, see, e.g., Zhang, Mykland and
Ait-Sahalia (2005), Bandi and Russell (2008), and
Barndorff-Nielsen, Hansen, Lunde and Shephard, henceforth BNHLS,
(2008). Return to text

3. Both studies explore the reliability of
the procedures, but only under relatively ideal circumstances and
not with a focus on the IQ estimator but rather the jump test
statistic. Return to text

4. We rely on the MedRV/MedRQ estimators
of Andersen, Dobrev and Schaumburg (2009) here, but any other
sensible pair of robust IV and IQ estimators would suffice in this
particular case as the evidence for a single significant jump is
compelling and the associated empirical inference problem thus very
straightforward. The robust estimators will be introduced later in
the paper. Return to text

5. As a consequence, real-time predictions
regarding the expected volatility over the following trading days
will likely also be exaggerated on 2/26/2008 because the jump
component of QV typically is much less persistent than the IV
component, see, e.g., Andersen, Bollerslev and Diebold
(2007). Return to text

6. Of course, the proofs for all our new
results, provided in Appendix A, allow for a general drift
term. Return to text

7. Naturally, additional dampening is
required to establish a CLT for the power variation measures. In
this case,
for
Return
to text

8. For IQ estimation based on a block
with five returns, we have
. Hence, the largest normalization constant exceeds the smallest
one by a factor of several hundred; see Table 1 for additional
information. Return to text

9. For the integrated variance, these
estimators may be interpreted as a redesigned version of the
original Quantile Realized Variation, or QRV, estimator of
Christensen, Oomen and Podolskij (COP) (2008), where it is applied
to quantiles based on absolute rather than raw returns. The
adaptation of QRV to cover the ADS (2012) nearest neighbor
truncation estimators is discussed in COP (2010). This was also
previously proposed during a conference discussion of COP (2010) by
Kevin Sheppard, see also the comments in the
realized_quantile_variance function in his Oxford MFE
Toolbox. Return to text

10. In practice, finite sample
complications can render the procedure less successful. One may
want to avoid estimators that are highly sensitive to
microstructure noise or jumps. Likewise, high correlation across
estimators can generate optimal combinations that are extreme which
may induce a degree of instability. Return to text

11. This is analogous to the use of linear
combinations for the QRV estimator explored by COP
(2008). Return to text

13. We intend to apply these estimators
in a setting with microstructure noise. As discussed in Section 4.2, this can be
accommodated in the finite activity jump scenario via pre-averaging
with a relatively large window size, resulting in a suboptimal
convergence rate, but added robustness to noise. The literature on
inference in the presence of infinite activity jumps and
noise is limited. However, the findings for MPV estimators in
Podolskij and Vetter (2009) and Hautsch and Podolskij (2010)
suggest constructive results are feasible. Establishing formal
results for this case falls outside the scope of the present
paper. Return to text

14. In this notation, the initial "R"
references the RNT estimator, "Min" ("Med") signals that we
exploit the smallest (median) realization of the underlying NT
estimators, and "RQ" ("RV") indicates realized quarticity
(realized variance) estimation, that is
(). It remains implicit that we take
, but retain only the three largest
absolute returns in constructing the estimator. Return to text

15. P1: Retain only observations with time
stamps between 9:30am and 4:00pm. P2: Retain only trades with
positive prices. P3: Retain only trades originating from the main
exchange (NYSE for all stocks except MSFT and INTL for which it is
NASDAQ). T1: Delete entries with corrected trades. T2: Delete
entries with abnormal sale condition. T3: If multiple trades
occurred with the same time stamp, use the median price. T4: Delete
entries with prices above the ask (or under the bid) by more than
one bid-ask spread. Return to
text

16. Analogous to the scaling factors
induced by the pre-averaging kernel, the efficiency with
subsampling depends on the pre-averaging scheme as well as the
length of the pre-averaging window relative to sample size. Hence,
applications involving subsampling must develop a suitable estimate
of the terms involving the efficiency factor in Section 3. One feasible
approach is simulation. Return to
text

18. We confirm that our filtering
procedure under the BM null, applied to pre-averaged returns, is
active for about 1-in- return blocks, while
the CPR filter is applied to roughly 1-in-
returns on average. Return to
text

19. These return intervals span the range
that is relevant for our empirical application. Findings based on a
more comprehensive set of frequencies are reported in ADS
(2011). Return to text

20. This choice facilitates effective
cancelation of noise and outliers across the numerator and
denominator. Return to text

21. Recall, we abandon the QRV style
estimators for IQ due to the rapidly deteriorating performance in
estimating higher order return variation under realistic market
conditions. Return to text

22. The improved performance for TBQ is
due to a fortuitous cancelation of the biases of the numerator and
denominator under the BM null. Return to
text

23. MPV and MPQ, in particular, as they
are based on products of adjacent (absolute) returns. Return to text

24. When using tick sampled data, we are
implicitly converting the calendar time scale to a tick scale,
where time evolves linearly in tick time. This implies that the
estimators are consistent for IQ in tick time, but not in calendar
time. Importantly, in this setting the (tick time) IQ represents
the relevant quantity for assessing the asymptotic variance of the
(tick time based) IV estimator and for inference regarding
jumps. Return to text

25. Ex post, we find no qualitative
differences in results between the two samples, so we only report
findings for the initial sample. The full set of results may be
gleaned from ADS (2011). Return to
text

26. Of course, this reflects the
different philosophy behind the mild functional truncation
filtering relative to TRQ and TBQ. The latter employ truncation of
single returns as the primary tool for achieving jump robustness,
and thus need to ensure that - asymptotically - all jumps are
prevented from impacting the IQ estimator. This requires aggressive
truncation and, inevitably, some truncation of diffusive returns,
motivating the CPR adjustment to mitigate the resulting
finite-sample bias, as discussed in Section 5. Return to text

27. We confirm that alternative robust
volatility-of-volatility measures lead to qualitatively similar
results. Return to text

28. The distance from the (non-functional
filtered) RV curve to the set of robust IV measures provides an
estimate of the average jump contribution to the quadratic return
variation. Return to text

29. For each individual robust estimator,
we regressed the difference of the daily realized power variation
estimate across the distinct frequencies against the daily VoV
measure, allowing for a separate intercept term but enforcing a
common slope across the stocks. The detailed results are available
in ADS (2011). Return to text

30. As argued in Barndorf-Nielsen,
Graversen, Podolskij, Jacod and Shephard (2006), henceforth BNGJPS,
this follows from working with the stopped versions of the
processes:
and
where
and
a.s. Return to text

31. As for the volatility process, the
specification may be generalized to infinite activity jump
processes along the lines of Barndorff-Nielsen et al
(2006c). Return to text

32. By Hölder's inequality,
, so that the distortion due to
microstructure noise in general will result in a downward bias in
the limiting ratio. Return to
text